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diff --git a/gae/n.lyx b/gae/n.lyx
new file mode 100644
index 0000000..ef7ae41
--- /dev/null
+++ b/gae/n.lyx
@@ -0,0 +1,212 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\usepackage{tikz}
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
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+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize a5paper
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 0.2cm
+\topmargin 0.7cm
+\rightmargin 0.2cm
+\bottommargin 0.7cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle empty
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Geometría afín y euclídea
+\end_layout
+
+\begin_layout Date
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+def
+\backslash
+cryear{2018}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "../license.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Bibliografía:
+\end_layout
+
+\begin_layout Itemize
+Material clases teóricas, Geometría Afín y Euclídea, Universidad de Murcia
+ (anónimo).
+\end_layout
+
+\begin_layout Chapter
+Espacios afines y variedades afines
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n1.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Aplicaciones afines
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n2.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Espacios euclídeos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Transformaciones ortogonales
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n4.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Movimientos
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n5.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/gae/n1.lyx b/gae/n1.lyx
new file mode 100644
index 0000000..acdf0f9
--- /dev/null
+++ b/gae/n1.lyx
@@ -0,0 +1,1753 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Espacios afines
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n1b.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Variedades afines
+\end_layout
+
+\begin_layout Standard
+Un subconjunto 
+\begin_inset Formula ${\cal L}\subseteq{\cal E}$
+\end_inset
+
+ es una 
+\series bold
+variedad (lineal) afín
+\series default
+ si 
+\begin_inset Formula $\exists P\in{\cal E},W\subseteq V:{\cal L}=P+W:=\{P+\vec{w}\}_{\vec{w}\in W}$
+\end_inset
+
+.
+ Se dice que 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ 
+\series bold
+pasa por
+\series default
+ el punto 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $W$
+\end_inset
+
+ es la 
+\series bold
+dirección
+\series default
+ de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ (
+\begin_inset Formula $\text{dir}({\cal L})=W$
+\end_inset
+
+), y se define la dimensión de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ como
+\begin_inset Formula 
+\[
+\dim({\cal L}):=\text{dim}(\text{dir}({\cal L}))=\dim_{K}(W)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una variedad de dimensión 1 es una 
+\series bold
+recta (afín)
+\series default
+, determinada por cualquier 
+\begin_inset Formula $P\in{\cal L}$
+\end_inset
+
+ y vector 
+\begin_inset Formula $\vec{v}\in\text{dir}({\cal L})$
+\end_inset
+
+ no nulo, llamado 
+\series bold
+vector director
+\series default
+ de la recta.
+ Una variedad de dimensión 2 es un 
+\series bold
+plano afín
+\series default
+, y una de dimensión 
+\begin_inset Formula $n-1$
+\end_inset
+
+ (con 
+\begin_inset Formula $n=\dim({\cal E})$
+\end_inset
+
+) es un 
+\series bold
+hiperplano afín
+\series default
+.
+ Así, para todo 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+, se tiene que 
+\begin_inset Formula $P+V={\cal E}$
+\end_inset
+
+.
+ Propiedades: Sean 
+\begin_inset Formula ${\cal L}=P+W$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}'=P'+W'$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $Q\in{\cal L}\iff\overrightarrow{PQ}\in W$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $Q\in{\cal L}\implies\exists\vec{w}\in W:Q=P+\vec{w}\implies\overrightarrow{PQ}=\vec{w}\in W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\overrightarrow{PQ}\in W\implies Q=P+\overrightarrow{PQ}\in P+W={\cal L}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $W=\{\overrightarrow{PR}\}_{R\in{\cal L}}=\{\overrightarrow{QR}\}_{Q,R\in{\cal L}}$
+\end_inset
+
+ (
+\begin_inset Formula $W$
+\end_inset
+
+ está unívocamente determinado por 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+).
+\begin_inset Newline newline
+\end_inset
+
+Vemos que 
+\begin_inset Formula $W\subseteq\{\overrightarrow{PR}\}_{R\in{\cal L}}\subseteq\{\overrightarrow{QR}\}_{Q,R\in{\cal L}}\subseteq W$
+\end_inset
+
+.
+ Primero, si 
+\begin_inset Formula $\vec{w}\in W$
+\end_inset
+
+, podemos definir 
+\begin_inset Formula $R:=P+\vec{w}\in{\cal L}$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\vec{w}=\overrightarrow{PR}\in\{\overrightarrow{PR}\}_{R\in{\cal L}}$
+\end_inset
+
+.
+ El segundo contenido es evidente, y para el tercero, dados 
+\begin_inset Formula $Q,R\in{\cal L}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\overrightarrow{PQ},\overrightarrow{PR}\in W$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\overrightarrow{QR}=\overrightarrow{PR}-\overrightarrow{PQ}\in W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P'\in{\cal L}\implies{\cal L}=P'+W$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sea 
+\begin_inset Formula ${\cal L}'=P'+W$
+\end_inset
+
+, como 
+\begin_inset Formula $P'\in{\cal L}$
+\end_inset
+
+, 
+\begin_inset Formula $\overrightarrow{PP'}\in W$
+\end_inset
+
+, y así,
+\begin_inset Formula 
+\[
+Q\in{\cal L}'\iff\overrightarrow{P'Q}\in W\iff\overrightarrow{PQ}=\overrightarrow{PP'}+\overrightarrow{P'Q}\in W\iff Q\in{\cal L}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $({\cal L},W,\varphi|_{{\cal L}\times W})$
+\end_inset
+
+ es un espacio afín.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula $Q\in{\cal L}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}\in W$
+\end_inset
+
+, entonces 
+\begin_inset Formula $Q+\vec{w}\in Q+W={\cal L}$
+\end_inset
+
+.
+ Las propiedades 
+\begin_inset Formula $(P+\vec{v})+\vec{w}=P+(\vec{v}+\vec{w})$
+\end_inset
+
+ y 
+\begin_inset Formula $P+\overrightarrow{0}=P$
+\end_inset
+
+ se cumplen trivialmente, y si 
+\begin_inset Formula $R,Q\in{\cal L}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\overrightarrow{RQ}\in W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}\subseteq{\cal L}'\iff W\subseteq W'\land\overrightarrow{PP'}\in W'\iff W\subseteq W'\land P\in{\cal L}'$
+\end_inset
+
+; 
+\begin_inset Formula ${\cal L}={\cal L}'\iff W=W'\land\overrightarrow{PP'}\in W$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Basta ver la primera serie de equivalencias.
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[1\implies2]$
+\end_inset
+
+ 
+\begin_inset Formula ${\cal L}\subseteq{\cal L}'\implies P\in{\cal L}'\implies\overrightarrow{PP'}\in W'$
+\end_inset
+
+.
+ Además, 
+\begin_inset Formula $W=\{\overrightarrow{QR}\}_{Q,R\in{\cal L}}\subseteq\{\overrightarrow{QR}\}_{Q,R\in{\cal L}'}=W'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[2\implies3]$
+\end_inset
+
+ 
+\begin_inset Formula $\overrightarrow{PP'}\in W'\implies\overrightarrow{P'P}\in W'\implies P\in{\cal L}'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[3\implies1]$
+\end_inset
+
+ 
+\begin_inset Formula $W\subseteq W'\land P\in{\cal L}'\implies{\cal L}=P+W\subseteq P+W'={\cal L}'$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Subsection
+Paralelismo, intersección y cruce de variedades
+\end_layout
+
+\begin_layout Standard
+Dos variedades 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}'$
+\end_inset
+
+ son 
+\series bold
+paralelas
+\series default
+ (
+\begin_inset Formula ${\cal L}\parallel{\cal L}'$
+\end_inset
+
+) si tienen la misma dirección.
+ Si solo se tiene que 
+\begin_inset Formula $\text{dir}({\cal L})\subseteq\text{dir}({\cal L}')$
+\end_inset
+
+, se dice que 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ es 
+\series bold
+débilmente paralela
+\series default
+ a 
+\begin_inset Formula ${\cal L}'$
+\end_inset
+
+ (
+\begin_inset Formula ${\cal L}\ll{\cal L}'$
+\end_inset
+
+).
+ Cuando no hay ambigüedad, a veces se omite el 
+\begin_inset Quotes cld
+\end_inset
+
+débilmente
+\begin_inset Quotes crd
+\end_inset
+
+.
+ Se trata de una relación reflexiva y transitiva en la que 
+\begin_inset Formula ${\cal L}\ll{\cal L}'\land{\cal L}'\ll{\cal L}\implies{\cal L}\parallel{\cal L}'$
+\end_inset
+
+, pero no es antisimétrica.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+postulado de las paralelas de Euclides
+\series default
+ afirma que por un punto exterior a una recta pasa una y sólo una paralela
+ a esta.
+ Esto se puede generalizar a que, dados 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ y una variedad afín 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+, existe una única variedad 
+\begin_inset Formula ${\cal L}'$
+\end_inset
+
+ que pasa por 
+\begin_inset Formula $P$
+\end_inset
+
+ y es paralela a 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+, y esta es 
+\begin_inset Formula ${\cal L}'=P+\text{dir}({\cal L})$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}\ll{\cal L}'\implies{\cal L}\subseteq{\cal L}'\lor{\cal L}\cap{\cal L}'=\emptyset$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+W\subseteq W'\land\exists Q\in{\cal L}\cap{\cal L}'\implies{\cal L}=Q+W\subseteq Q+W'={\cal L}'
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}\parallel{\cal L}'\implies{\cal L}={\cal L}'\lor{\cal L}\cap{\cal L}'=\emptyset$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+W=W'\land\exists Q\in{\cal L}\cap{\cal L}'\implies{\cal L}=Q+W=Q+W'={\cal L}'
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}\ll{\cal L}'\iff\exists{\cal S}:{\cal L}\parallel{\cal S}\subseteq{\cal L}'\iff\exists{\cal S}':{\cal L}\subseteq{\cal S}'\parallel{\cal L}'$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[1\implies2,3]$
+\end_inset
+
+ 
+\begin_inset Formula $W\subseteq W'\implies{\cal L}=P+W\parallel P'+W\subseteq P'+W'={\cal L}'\land{\cal L}=P+W\subseteq P+W'\parallel P'+W'={\cal L}'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[2\implies1]$
+\end_inset
+
+ 
+\begin_inset Formula ${\cal L}\parallel{\cal S}\subseteq{\cal L}'\implies\text{dir}({\cal L})=\text{dir}({\cal S})\subseteq\text{dir}({\cal L}')\implies{\cal L}\ll{\cal L}'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[3\implies1]$
+\end_inset
+
+ 
+\begin_inset Formula ${\cal L}\subseteq{\cal S}'\parallel{\cal L}\implies\text{dir}({\cal L})\subseteq\text{dir}({\cal S}')=\text{dir}({\cal L}')\implies{\cal L}\ll{\cal L}'$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Se dice que dos variedades 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}'$
+\end_inset
+
+ 
+\series bold
+se cortan
+\series default
+ o son 
+\series bold
+incidentes
+\series default
+ si 
+\begin_inset Formula ${\cal L}\cap{\cal L}'\neq\emptyset$
+\end_inset
+
+, y que 
+\series bold
+se cruzan
+\series default
+ si ni se cortan ni ninguna es débilmente paralela a la otra.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\{{\cal L}_{i}\}_{i\in I}$
+\end_inset
+
+ es una familia de variedades afines de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ con 
+\begin_inset Formula ${\cal L}_{i}=P+W_{i}\forall i\in I$
+\end_inset
+
+ y 
+\begin_inset Formula $\bigcap_{i\in I}{\cal L}_{i}\neq\emptyset$
+\end_inset
+
+ entonces la intersección es una variedad afín con dirección 
+\begin_inset Formula $\bigcap_{i\in I}W_{i}$
+\end_inset
+
+.
+ 
+\begin_inset Formula 
+\[
+Q\in P+\bigcap_{i\in I}W_{i}\iff\forall i\in I,\overrightarrow{PQ}\in W_{i}\iff\forall i\in I,Q\in P+W_{i}={\cal L}_{i}\iff Q\in\bigcap_{i\in I}{\cal L}_{i}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}\cap{\cal L}'\neq\emptyset\iff\overrightarrow{PP'}\in W+W'$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $Q\in{\cal L}\cap{\cal L}'\implies\overrightarrow{PP'}=\overrightarrow{PQ}+\overrightarrow{QP'}\in W+W'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\exists\vec{w}\in W,\vec{w}'\in W':\overrightarrow{PP'}=\vec{w}+\vec{w}'\implies P+\vec{w}=P+\overrightarrow{PP'}-\vec{w}'=P'-\vec{w}'\in{\cal L}\cap{\cal L}'$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Dos variedades 
+\begin_inset Formula ${\cal L}=P+W$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}'=P'+W'$
+\end_inset
+
+ son 
+\series bold
+complementarias
+\series default
+ si lo son sus direcciones, es decir, si 
+\begin_inset Formula $V=W\oplus W'$
+\end_inset
+
+.
+ La intersección de dos variedades afines complementarias es un punto.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $\overrightarrow{PP'}\in V=W\oplus W'$
+\end_inset
+
+, luego se cortan, y 
+\begin_inset Formula $W\cap W'=\{0\}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\dim({\cal L}\cap{\cal L}')=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Suma de variedades
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+variedad afín engendrada
+\series default
+ o 
+\series bold
+generada
+\series default
+ por 
+\begin_inset Formula $X\subseteq{\cal E}$
+\end_inset
+
+ a la menor de las variedades que contienen a 
+\begin_inset Formula $X$
+\end_inset
+
+, es decir, la intersección de todas ellas, y se denota por 
+\begin_inset Formula ${\cal V}(X)$
+\end_inset
+
+.
+ Esta existe porque la intersección no es vacía (contiene a 
+\begin_inset Formula $X$
+\end_inset
+
+) y al menos 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ es una variedad que contiene a 
+\begin_inset Formula $X$
+\end_inset
+
+.
+ Dados 
+\begin_inset Formula $P_{1},\dots,P_{n}\in{\cal E}$
+\end_inset
+
+, se tiene que 
+\begin_inset Formula ${\cal V}(P_{1},\dots,P_{n})=P_{1}+<\overrightarrow{P_{1}P_{2}},\dots,\overrightarrow{P_{1}P_{n}}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $P_{1}+<\overrightarrow{P_{1}P_{2}},\dots,\overrightarrow{P_{1}P_{n}}>$
+\end_inset
+
+ contiene a 
+\begin_inset Formula $P_{1},P_{2},\dots,P_{n}$
+\end_inset
+
+, luego contiene a 
+\begin_inset Formula ${\cal V}(X)$
+\end_inset
+
+ por ser una de las variedades que se intersecan.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula ${\cal V}(P_{1},\dots,P_{n})$
+\end_inset
+
+ pasa por 
+\begin_inset Formula $P_{1}$
+\end_inset
+
+ y su dirección debe contener a los 
+\begin_inset Formula $\overrightarrow{P_{1}P_{j}}$
+\end_inset
+
+ (
+\begin_inset Formula $2\leq j\leq n$
+\end_inset
+
+) y por tanto a 
+\begin_inset Formula $<\overrightarrow{P_{1}P_{2}},\dots,\overrightarrow{P_{1}P_{n}}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+suma
+\series default
+ de 
+\begin_inset Formula $\{{\cal L}_{i}\}_{i\in I}$
+\end_inset
+
+ es la variedad engendrada por su unión: 
+\begin_inset Formula $\sum_{i\in I}{\cal L}_{i}:={\cal V}\left(\bigcup_{i\in I}{\cal L}_{i}\right)$
+\end_inset
+
+.
+ Se tiene que dadas 
+\begin_inset Formula ${\cal L}=P+W$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}'=P'+W'$
+\end_inset
+
+, entonces 
+\begin_inset Formula ${\cal L}+{\cal L}'=P+(W+W'+<\overrightarrow{PP'}>)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+La variedad a la derecha del igual contiene a 
+\begin_inset Formula $P+W={\cal L}$
+\end_inset
+
+, y como en esta podemos cambiar 
+\begin_inset Formula $P$
+\end_inset
+
+ por 
+\begin_inset Formula $P'=P+\overrightarrow{PP'}$
+\end_inset
+
+, también contiene a 
+\begin_inset Formula $P'+W'={\cal L}'$
+\end_inset
+
+, luego contiene a la suma.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Evidentemente, 
+\begin_inset Formula $P\in{\cal L}+{\cal L}'$
+\end_inset
+
+.
+ Ahora bien, como 
+\begin_inset Formula ${\cal L},{\cal L}'\subseteq{\cal L}+{\cal L}'$
+\end_inset
+
+, entonces 
+\begin_inset Formula $W,W'\subseteq\text{dir}({\cal L}+{\cal L}')$
+\end_inset
+
+, y como 
+\begin_inset Formula $P,P'\in{\cal L}+{\cal L}'$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\overrightarrow{PP'}\in\text{dir}({\cal L}+{\cal L}')$
+\end_inset
+
+, luego 
+\begin_inset Formula $W+W'+<\overrightarrow{PP'}>\subseteq\text{dir}({\cal L}+{\cal L}')$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Fórmulas de Grassmann:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}\cap{\cal L}'\neq\emptyset\implies\dim({\cal L}+{\cal L}')=\dim({\cal L})+\dim({\cal L}')-\dim({\cal L}\cap{\cal L}')$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+En este caso, 
+\begin_inset Formula $\text{dir}({\cal L}\cap{\cal L}')=W\cap W'$
+\end_inset
+
+, y como 
+\begin_inset Formula $\overrightarrow{PP'}\in W+W'$
+\end_inset
+
+, entonces
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $W+W'+$
+\end_inset
+
+
+\begin_inset Formula $<\overrightarrow{PP'}>=W+W'$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\begin{array}{c}
+\dim({\cal L}+{\cal L}')=\dim(W+W')=\dim(W)+\dim(W')-\dim(W\cap W')=\\
+=\dim({\cal L})+\dim({\cal L}')-\dim({\cal L}\cap{\cal L}')
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}\cap{\cal L}'=\emptyset\implies\dim({\cal L}+{\cal L}')=\dim({\cal L})+\dim({\cal L}')-\dim(W\cap W')+1$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+En este caso, 
+\begin_inset Formula $\overrightarrow{PP'}\notin W+W'$
+\end_inset
+
+, por lo que
+\begin_inset Formula 
+\[
+\begin{array}{c}
+\dim({\cal L}+{\cal L}')=\dim(W+W'+\overrightarrow{PP'})=\dim(W+W')+1=\\
+=\dim(W)+\dim(W')-\dim(W\cap W')+1
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Posición relativa de variedades
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula ${\cal L}_{i}=P_{i}+<\vec{v}_{i}>$
+\end_inset
+
+ (
+\begin_inset Formula $i\in\{1,2\},\vec{v}_{i}\neq\vec{0}$
+\end_inset
+
+) dos rectas en un plano afín.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\vec{v}_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v}_{2}$
+\end_inset
+
+ son proporcionales entonces 
+\begin_inset Formula ${\cal L}_{1}\parallel{\cal L}_{2}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\overrightarrow{P_{1}P_{2}}\in<\vec{v}_{1}>$
+\end_inset
+
+, son coincidentes; en otro caso son paralelas distintas.
+\end_layout
+
+\begin_layout Itemize
+En otro caso son subespacios complementarios y por tanto se cortan en un
+ punto.
+\end_layout
+
+\begin_layout Standard
+Si tenemos dos rectas en un espacio tridimensional, la discusión es similar
+ a cuando estamos en el plano afín, pero si las rectas no son paralelas,
+ sólo se cortan si 
+\begin_inset Formula $\overrightarrow{P_{1}P_{2}}\in<\vec{v}_{1},\vec{v}_{2}>$
+\end_inset
+
+, de lo contrario se cruzan.
+ Sean ahora tres rectas, sin ser dos de ellas coincidentes, en un plano
+ afín.
+\end_layout
+
+\begin_layout Itemize
+Si hay dos paralelas, digamos 
+\begin_inset Formula ${\cal L}_{1}\parallel{\cal L}_{2}$
+\end_inset
+
+, si 
+\begin_inset Formula $\vec{v}_{3}$
+\end_inset
+
+ es proporcional a 
+\begin_inset Formula $\vec{v}_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v}_{2}$
+\end_inset
+
+ tenemos tres paralelas distintas, de lo contrario 
+\begin_inset Formula ${\cal L}_{3}$
+\end_inset
+
+ corta en un punto a cada una de las otras.
+\end_layout
+
+\begin_layout Itemize
+En otro caso, cada par de rectas se cortan en un punto.
+ Si dos de estos coinciden, también coinciden con el tercero, y de lo contrario
+ las rectas se cortan en puntos distintos dos a dos.
+\end_layout
+
+\begin_layout Standard
+Ahora, sean 
+\begin_inset Formula ${\cal L}=P+<\vec{v}>$
+\end_inset
+
+ (
+\begin_inset Formula $\vec{v}\neq\vec{0}$
+\end_inset
+
+) y 
+\begin_inset Formula ${\cal P}=P'+W$
+\end_inset
+
+ una recta y plano en un espacio afín tridimensional:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\vec{v}\in W$
+\end_inset
+
+ entonces 
+\begin_inset Formula ${\cal L}\ll{\cal P}$
+\end_inset
+
+, y en particular, si 
+\begin_inset Formula $P\in{\cal P}$
+\end_inset
+
+ entonces 
+\begin_inset Formula ${\cal L}\subseteq{\cal P}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\vec{v}\notin W$
+\end_inset
+
+, las variedades son complementarias, luego se cortan en un punto.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula ${\cal P}_{i}=P_{i}+W_{i}$
+\end_inset
+
+ (
+\begin_inset Formula $i\in\{1,2\},\dim(W_{i})=2$
+\end_inset
+
+) dos planos en un espacio afín tridimensional.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $W_{1}=W_{2}$
+\end_inset
+
+, los planos son paralelos.
+ En particular, son coincidentes si 
+\begin_inset Formula $\overrightarrow{P_{1}P_{2}}\in W_{1}$
+\end_inset
+
+; de lo contrario son paralelos distintos.
+\end_layout
+
+\begin_layout Itemize
+En otro caso, se tiene que 
+\begin_inset Formula $\dim(W_{1}\cap W_{2})=1$
+\end_inset
+
+ y 
+\begin_inset Formula $\dim(W_{1}+W_{2})=3$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\overrightarrow{P_{1}P_{2}}\in W_{1}+W_{2}$
+\end_inset
+
+ y los planos se cortan en una recta de dirección 
+\begin_inset Formula $W_{1}\cap W_{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si ahora consideramos tres planos ninguno coincidente con ningún otro, entonces:
+\end_layout
+
+\begin_layout Itemize
+Si hay dos paralelos, digamos 
+\begin_inset Formula ${\cal P}_{1}\parallel{\cal P}_{2}$
+\end_inset
+
+, si 
+\begin_inset Formula $W_{3}=W_{1}$
+\end_inset
+
+ tenemos tres planos paralelos distintos; de lo contrario 
+\begin_inset Formula ${\cal P}_{3}$
+\end_inset
+
+ corta en una recta a cada uno de los otros.
+\end_layout
+
+\begin_layout Itemize
+En otro caso, sea 
+\begin_inset Formula ${\cal L}={\cal P}_{1}\cap{\cal P}_{2}=P+W\neq\emptyset$
+\end_inset
+
+, si 
+\begin_inset Formula ${\cal L}\subseteq{\cal P}_{3}$
+\end_inset
+
+, entonces 
+\begin_inset Formula ${\cal L}={\cal P}_{1}\cap{\cal P}_{2}\cap{\cal P}_{3}$
+\end_inset
+
+ y los tres planos se cortan en una recta.
+ Si 
+\begin_inset Formula ${\cal L}\ll{\cal P}_{3}$
+\end_inset
+
+ (
+\begin_inset Formula $W\subseteq W_{3}$
+\end_inset
+
+) entonces 
+\begin_inset Formula $W\subseteq W_{1}\cap W_{3}$
+\end_inset
+
+, y como 
+\begin_inset Formula $\dim(W_{1}\cap W_{3})=\dim(W)=1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $W=W_{1}\cap W_{3}$
+\end_inset
+
+ y del mismo modo 
+\begin_inset Formula $W=W_{2}\cap W_{3}$
+\end_inset
+
+, luego los planos se cortan dos a dos en paralelas distintas.
+ Finalmente, si 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal P}_{3}$
+\end_inset
+
+ se cortan en un punto, los tres planos se cortan en este.
+\end_layout
+
+\begin_layout Section
+Ecuaciones de variedades afines
+\end_layout
+
+\begin_layout Standard
+En esta sección asumimos 
+\begin_inset Formula $\dim({\cal E})=n$
+\end_inset
+
+ e identificamos los vectores con sus coordenadas en 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ y los puntos con sus coordenadas en 
+\begin_inset Formula $\Re:=(O,{\cal B})$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula ${\cal L}=P+W$
+\end_inset
+
+ con 
+\begin_inset Formula $W=<\vec{v}_{1},\dots,\vec{v}_{m}>$
+\end_inset
+
+, los puntos de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ tienen la forma 
+\begin_inset Formula $X=P+\lambda_{1}\vec{v}_{1}+\dots+\lambda_{m}\vec{v}_{m}$
+\end_inset
+
+, con cada 
+\begin_inset Formula $\lambda_{i}\in K$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $[X]_{\Re}=(x_{1},\dots,x_{n})$
+\end_inset
+
+, 
+\begin_inset Formula $[P]_{\Re}=(p_{1},\dots,p_{n})$
+\end_inset
+
+ y 
+\begin_inset Formula $[\vec{v}_{i}]_{{\cal B}}=(v_{1i},\dots,v_{ni})$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{ccc}
+x_{1} & = & p_{1}+\lambda_{1}v_{11}+\dots+\lambda_{m}v_{1m}\\
+ & \vdots\\
+x_{n} & = & p_{n}+\lambda_{1}v_{n1}+\dots+\lambda_{m}v_{nm}
+\end{array}\right.
+\]
+
+\end_inset
+
+Estas son las 
+\series bold
+ecuaciones paramétricas
+\series default
+ de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ en 
+\begin_inset Formula $\Re$
+\end_inset
+
+, y no son únicas.
+ Si 
+\begin_inset Formula $\vec{v}_{1},\dots,\vec{v}_{m}$
+\end_inset
+
+ son li
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+ne
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+al
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+men
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+te independientes entonces el número de parámetros es la dimensión de 
+\begin_inset Formula $W$
+\end_inset
+
+ y de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $W$
+\end_inset
+
+ viene dado por ecuaciones cartesianas en 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ representadas por un sistema homogéneo con matriz de coeficientes 
+\begin_inset Formula $A$
+\end_inset
+
+, es decir, si 
+\begin_inset Formula $\vec{v}\in W\iff A\vec{v}=0$
+\end_inset
+
+, entonces 
+\begin_inset Formula $X\in{\cal L}\iff\overrightarrow{PX}\in W\iff A(X-P)=0\iff AX=AP$
+\end_inset
+
+.
+ El resultado es un sistema de ecuaciones, denominadas 
+\series bold
+ecuaciones cartesianas
+\series default
+ o 
+\series bold
+implícitas
+\series default
+ de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ en 
+\begin_inset Formula $\Re$
+\end_inset
+
+, que no es único, y cuyas soluciones son los puntos de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $r=\text{rg}A$
+\end_inset
+
+ (el rango del sistema), entonces 
+\begin_inset Formula $\dim({\cal L})=\dim({\cal E})-r$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para obtener las paramétricas (o las implícitas) de 
+\begin_inset Formula $W$
+\end_inset
+
+ a partir de las correspondientes de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+, basta anular los términos independientes en cada caso.
+ Así, para obtener las paramétricas de la recta paralela a 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ por 
+\begin_inset Formula $P'$
+\end_inset
+
+, basta sustituir las coordenadas de 
+\begin_inset Formula $P$
+\end_inset
+
+ (
+\begin_inset Formula $p_{1},\dots,p_{n}$
+\end_inset
+
+) por las de 
+\begin_inset Formula $P'$
+\end_inset
+
+en las paramétricas de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+.
+ Para obtener las implícitas, si las de 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ son 
+\begin_inset Formula $\left(\begin{array}{c|c}
+A & B\end{array}\right)$
+\end_inset
+
+, las de la paralela son 
+\begin_inset Formula $\left(\begin{array}{c|c}
+A & AP'\end{array}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para obtener ecuaciones paramétricas a partir de implícitas, resolvemos
+ el sistema 
+\begin_inset Formula $(A|B)$
+\end_inset
+
+ en función de parámetros, y para pasar de paramétricas a implícitas (por
+ ejemplo, el sistema de arriba), consideramos la matriz
+\begin_inset Formula 
+\[
+\left(\begin{array}{ccc|c}
+v_{11} & \cdots & v_{1m} & x_{1}-p_{1}\\
+\vdots &  & \vdots & \vdots\\
+v_{n1} & \cdots & v_{nm} & x_{n}-p_{n}
+\end{array}\right)
+\]
+
+\end_inset
+
+y se trata de discutir el sistema que forma.
+ Lo mejor en general es hacerlo por menores, pues si los 
+\begin_inset Formula $m$
+\end_inset
+
+ vectores iniciales son linealmente independientes, el rango de la matriz
+ debe ser 
+\begin_inset Formula $m$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para obtener la intersección de dos variedades dadas sus ecuaciones implícitas,
+ basta juntarlas.
+ También, si conocemos las implícitas de una y las paramétricas de la segunda,
+ podemos sustituir el 
+\begin_inset Quotes cld
+\end_inset
+
+punto genérico
+\begin_inset Quotes crd
+\end_inset
+
+ que nos dan las paramétricas de la segunda y sustituirlo en la primera,
+ obteniendo como resultado las condiciones para que un punto de la segunda
+ esté además en la primera.
+ Por otro lado, si tenemos las paramétricas de dos variedades y queremos
+ hallar su suma, basta recordar que 
+\begin_inset Formula ${\cal L}+{\cal L}'=P+(W+W'+<\overrightarrow{PP'}>)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Ejemplos en dimensiones bajas
+\end_layout
+
+\begin_layout Standard
+Una recta en un plano afín es un hiperplano, por lo que viene dada por una
+ sóla ecuación 
+\begin_inset Formula 
+\[
+\left|\begin{array}{cc}
+v_{1} & x_{1}-p_{1}\\
+v_{2} & x_{2}-p_{2}
+\end{array}\right|=0
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $(p_{1},p_{2})\neq(q_{1},q_{2})$
+\end_inset
+
+, la recta que los une tiene como ecuación
+\begin_inset Formula 
+\[
+\left|\begin{array}{cc}
+q_{1}-p_{1} & x_{1}-p_{1}\\
+q_{2}-p_{2} & x_{2}-p_{2}
+\end{array}\right|=\left|\begin{array}{ccc}
+1 & 1 & 1\\
+p_{1} & q_{1} & x_{1}\\
+p_{2} & q_{2} & x_{2}
+\end{array}\right|=0
+\]
+
+\end_inset
+
+lo que sirve para comprobar si tres puntos están alineados.
+ Decimos que unos puntos son 
+\series bold
+coplanarios
+\series default
+ si existe un plano que los contiene a todos.
+ Los planos en un espacio tridimensional son hiperplanos, y su ecuación
+ implícita es
+\begin_inset Formula 
+\[
+\left|\begin{array}{ccc}
+v_{1} & w_{1} & x_{1}-p_{1}\\
+v_{2} & w_{2} & x_{2}-p_{2}\\
+v_{3} & w_{3} & x_{3}-p_{3}
+\end{array}\right|=0
+\]
+
+\end_inset
+
+Así, si tres puntos 
+\begin_inset Formula $P$
+\end_inset
+
+, 
+\begin_inset Formula $Q$
+\end_inset
+
+ y 
+\begin_inset Formula $R$
+\end_inset
+
+ no están alineados, forman un plano dado por
+\begin_inset Formula 
+\[
+\left|\begin{array}{ccc}
+q_{1}-p_{1} & r_{1}-p_{1} & x_{1}-p_{1}\\
+q_{2}-p_{2} & r_{2}-p_{2} & x_{2}-p_{2}\\
+q_{3}-p_{3} & r_{3}-p_{3} & x_{3}-p_{3}
+\end{array}\right|=\left|\begin{array}{cccc}
+1 & 1 & 1 & 1\\
+p_{1} & q_{1} & r_{1} & s_{1}\\
+p_{2} & q_{2} & r_{2} & s_{2}\\
+p_{3} & q_{3} & r_{3} & s_{3}
+\end{array}\right|=0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En un espacio tridimensional, el punto 
+\begin_inset Formula $(x_{1},x_{2},x_{3})$
+\end_inset
+
+ está en la recta
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\ell=(p_{1},p_{2},p_{3})+$
+\end_inset
+
+
+\begin_inset Formula $<(v_{1},v_{2},v_{3})>$
+\end_inset
+
+ cuando 
+\begin_inset Formula $(v_{1},v_{2},v_{3})$
+\end_inset
+
+ y 
+\begin_inset Formula $(x_{1}-p_{1},x_{2}-p_{2},x_{3}-p_{3})$
+\end_inset
+
+ sean proporcionales, lo que nos lleva a las 
+\series bold
+ecuaciones continuas
+\series default
+:
+\begin_inset Formula 
+\[
+\frac{x_{1}-p_{1}}{v_{1}}=\frac{x_{2}-p_{2}}{v_{2}}=\frac{x_{3}-p_{3}}{v_{3}}
+\]
+
+\end_inset
+
+Si una de las coordenadas del vector director es 0, este caso debe ser tratado
+ de forma especial.
+ A partir de estas ecuaciones podemos obtener las implícitas.
+ El 
+\series bold
+haz de planos
+\series default
+ que contienen a 
+\begin_inset Formula $\ell$
+\end_inset
+
+ es el conjunto de todos los planos que la contienen.
+ Así, si
+\begin_inset Formula 
+\[
+\ell\equiv\left\{ \begin{array}{rcl}
+ax+by+cz+d & = & 0\\
+a'x+b'y+c'z+d' & = & 0
+\end{array}\right.
+\]
+
+\end_inset
+
+su haz de planos está formado por las combinaciones lineales de estas ecuaciones
+, es decir, el plano 
+\begin_inset Formula $a'x+b'y+c'z+d'=0$
+\end_inset
+
+ y los planos 
+\begin_inset Formula $(ax+by+cz+d)+\mu(a'x+b'y+c'z+d)=0$
+\end_inset
+
+ con 
+\begin_inset Formula $\mu\in K$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/gae/n1b.lyx b/gae/n1b.lyx
new file mode 100644
index 0000000..7f06d5c
--- /dev/null
+++ b/gae/n1b.lyx
@@ -0,0 +1,1010 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\begin_preamble
+\input defs
+\end_preamble
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+A lo largo del capítulo, cuando no haya ambigüedad, identificamos el espacio
+ afín 
+\begin_inset Formula $({\cal E},V,\varphi)$
+\end_inset
+
+ con el conjunto 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+.
+ Un 
+\series bold
+espacio afín
+\series default
+ sobre un cuerpo 
+\begin_inset Formula $K$
+\end_inset
+
+ es una terna 
+\begin_inset Formula $({\cal E},V,\varphi)$
+\end_inset
+
+ formada por un conjunto 
+\begin_inset Formula ${\cal E}\neq0$
+\end_inset
+
+, cuyos elementos llamamos 
+\series bold
+puntos
+\series default
+; un 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $V$
+\end_inset
+
+, llamado 
+\series bold
+espacio vectorial asociado
+\series default
+ a o 
+\series bold
+de direcciones
+\series default
+ de 
+\begin_inset Formula $({\cal E},V,\varphi)$
+\end_inset
+
+, y una aplicación 
+\begin_inset Formula $\varphi:{\cal E}\times V\rightarrow{\cal E}$
+\end_inset
+
+, que escribimos como 
+\begin_inset Formula $P+\vec{v}:=\varphi(P,\vec{v})$
+\end_inset
+
+, que cumplen que 
+\begin_inset Formula $\forall P,Q\in{\cal E},\vec{v},\vec{w}\in V$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $(P+\vec{v})+\vec{w}=P+(\vec{v}+\vec{w})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P+\vec{0}=P$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\exists!\overrightarrow{PQ}\in V:P+\overrightarrow{PQ}=Q$
+\end_inset
+
+.
+ Decimos que 
+\begin_inset Formula $P$
+\end_inset
+
+ es el 
+\series bold
+origen
+\series default
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ el 
+\series bold
+extremo
+\series default
+ del vector 
+\begin_inset Formula $\overrightarrow{PQ}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\overrightarrow{P(P+\vec{v})}=\vec{v}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+dimensión
+\series default
+ de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ a la de su espacio vectorial asociado, 
+\begin_inset Formula $\dim({\cal E})=\dim_{K}(V)$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+rectas afines
+\series default
+ a los espacios afines de dimensión 1, 
+\series bold
+planos afines
+\series default
+ a los de dimensión 2 y 
+\series bold
+espacios (tridimensionales) afines
+\series default
+ a los de dimensión 3.
+\end_layout
+
+\begin_layout Standard
+Tenemos que, dado 
+\begin_inset Formula $O\in{\cal E}$
+\end_inset
+
+, las aplicaciones 
+\begin_inset Formula $V\rightarrow{\cal E}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal E}\rightarrow V$
+\end_inset
+
+ dadas, respectivamente, por 
+\begin_inset Formula $\vec{v}\mapsto O+\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $P\mapsto\overrightarrow{OP}$
+\end_inset
+
+ son biyecciones una inversa de la otra.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $\vec{v}\mapsto O+\vec{v}\mapsto\overrightarrow{O(O+\vec{v})}=\vec{v}$
+\end_inset
+
+; 
+\begin_inset Formula $P\mapsto\overrightarrow{OP}\mapsto O+\overrightarrow{OP}=P$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Esta biyección permite dar a 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ una estructura de espacio vectorial definida por 
+\begin_inset Formula $P\hat{+}Q=O+(\overrightarrow{OP}+\overrightarrow{OQ})$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\cdot P=O+\lambda\overrightarrow{OP}$
+\end_inset
+
+, a la que llamamos 
+\series bold
+vectorialización
+\series default
+ de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ respecto a 
+\begin_inset Formula $O\in{\cal E}$
+\end_inset
+
+, que es isomorfa a 
+\begin_inset Formula $V$
+\end_inset
+
+ y cuyo elemento neutro es 
+\begin_inset Formula $O$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Algunos espacios afines:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Espacio afín trivial:
+\series default
+ De dimensión 0, con un solo punto, pues dados 
+\begin_inset Formula $P,Q\in{\cal E}$
+\end_inset
+
+, 
+\begin_inset Formula $Q=P+\overrightarrow{PQ}=P+\vec{0}=P$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Estructura afín de un espacio vectorial:
+\series default
+ Dado un 
+\begin_inset Formula $K$
+\end_inset
+
+-espacio vectorial 
+\begin_inset Formula $V$
+\end_inset
+
+, existe un espacio afín 
+\begin_inset Formula $(V,V,\varphi)$
+\end_inset
+
+ donde la suma es la suma usual de vectores.
+ Podemos entonces escribir 
+\begin_inset Formula $\overrightarrow{PQ}=Q-P$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+espacio afín numérico
+\series default
+ de dimensión 
+\begin_inset Formula $n$
+\end_inset
+
+ sobre 
+\begin_inset Formula $K$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal E}^{n}(K)$
+\end_inset
+
+, a la estructura afín de 
+\begin_inset Formula $K^{n}$
+\end_inset
+
+.
+ 
+\begin_inset Formula ${\cal E}^{2}(\mathbb{R})$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal E}^{3}(\mathbb{R})$
+\end_inset
+
+ son pues el plano y el espacio afín usuales.
+\end_layout
+
+\begin_layout Subsection
+Propiedades
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overrightarrow{PQ}=\vec{0}\iff P=Q$
+\end_inset
+
+; 
+\begin_inset Formula $\overrightarrow{PP}=\vec{0}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\begin{array}{c}
+\overrightarrow{PQ}=\vec{0}\implies Q=P+\overrightarrow{PQ}=P+\vec{0}=P\\
+Q+\vec{0}=Q\implies\overrightarrow{QQ}=\vec{0}
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Relación de Chasles:
+\series default
+ 
+\begin_inset Formula $\overrightarrow{P_{1}P_{2}}+\overrightarrow{P_{2}P_{3}}+\dots+\overrightarrow{P_{n-1}P_{n}}=\overrightarrow{P_{1}P_{n}}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+P+(\overrightarrow{PQ}+\overrightarrow{QR})=(P+\overrightarrow{PQ})+\overrightarrow{QR}=Q+\overrightarrow{QR}=R
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overrightarrow{PQ}=-\overrightarrow{QP}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\overrightarrow{PQ}+\overrightarrow{QP}=\overrightarrow{PP}=\overrightarrow{0}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Cancelación:
+\series default
+ 
+\begin_inset Formula $P+\vec{v}=P+\vec{w}\implies\vec{v}=\vec{w}$
+\end_inset
+
+; 
+\begin_inset Formula $P+\vec{v}=Q+\vec{v}\implies P=Q$
+\end_inset
+
+; 
+\begin_inset Formula $\overrightarrow{PQ}=\overrightarrow{PR}\iff Q=R\iff\overrightarrow{QP}=\overrightarrow{RP}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\begin{array}{c}
+P+\vec{v}=P+\vec{w}\implies\vec{v}=\overrightarrow{P(P+\vec{v})}=\overrightarrow{P(P+\vec{w})}=\vec{w}\\
+P+\vec{v}=Q+\vec{v}\implies P=P+\vec{v}-\vec{v}=Q+\vec{v}-\vec{v}=Q\\
+\overrightarrow{PQ}=\overrightarrow{PR}\implies Q=P+\overrightarrow{PQ}=P+\overrightarrow{PR}=R
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\overrightarrow{(P+\vec{v})(Q+\vec{w})}=\overrightarrow{PQ}+\vec{w}-\vec{v}$
+\end_inset
+
+; 
+\begin_inset Formula $\overrightarrow{P(Q+\vec{w})}=\overrightarrow{PQ}+\vec{w}$
+\end_inset
+
+; 
+\begin_inset Formula $\overrightarrow{(P+\vec{v})Q}=\overrightarrow{PQ}-\vec{v}$
+\end_inset
+
+; 
+\begin_inset Formula $\overrightarrow{(P+\vec{v})P}=-\vec{v}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+(P+\vec{v})+(\overrightarrow{PQ}+\vec{w}-\vec{v})=P+\overrightarrow{PQ}+\vec{w}=Q+\vec{w}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $P+\vec{v}=Q+\vec{w}\iff\overrightarrow{PQ}=\vec{v}-\vec{w}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+P+\vec{v}=Q+\vec{w}\iff\overrightarrow{(P+\vec{v})(Q+\vec{w})}=\overrightarrow{PQ}+\vec{w}-\vec{v}=\overrightarrow{0}\iff\overrightarrow{PQ}=\vec{w}-\vec{v}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Regla del paralelogramo:
+\series default
+ 
+\begin_inset Formula $\overrightarrow{PQ}=\overrightarrow{P'Q'}\iff\overrightarrow{PP'}=\overrightarrow{QQ'}$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\overrightarrow{PQ}+\overrightarrow{QQ'}=\overrightarrow{PQ'}=\overrightarrow{PP'}+\overrightarrow{P'Q'}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Sistemas de referencia y coordenadas
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+sistema de referencia
+\series default
+ (o 
+\series bold
+referencial
+\series default
+)
+\series bold
+ cartesiano
+\series default
+ de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ es un par 
+\begin_inset Formula $\Re=(O,{\cal B})$
+\end_inset
+
+ formado por un 
+\series bold
+origen
+\series default
+ 
+\begin_inset Formula $O\in{\cal E}$
+\end_inset
+
+ y una base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Las 
+\series bold
+coordenadas (cartesianas)
+\series default
+ de 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ en 
+\begin_inset Formula $\Re$
+\end_inset
+
+ son las del vector 
+\begin_inset Formula $\overrightarrow{OP}$
+\end_inset
+
+ respecto de la base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+, y se denotan 
+\begin_inset Formula $[P]_{\Re}:=[\overrightarrow{OP}]_{{\cal B}}$
+\end_inset
+
+.
+ En particular 
+\begin_inset Formula $[O]_{\Re}=(0,\dots,0)$
+\end_inset
+
+, 
+\begin_inset Formula $[P+\vec{v}]_{\Re}=[P]_{\Re}+[\vec{v}]_{{\cal B}}$
+\end_inset
+
+ y 
+\begin_inset Formula $[\overrightarrow{PQ}]_{{\cal B}}=[Q]_{\Re}-[P]_{\Re}$
+\end_inset
+
+.
+ Cuando se trabaja con un único referencial, se omiten los subíndices 
+\begin_inset Formula $\Re$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ en los corchetes, o incluso se pueden identificar los puntos y vectores
+ con sus coordenadas, siempre que se indique esto al principio de trabajar
+ con coordenadas, y podemos entonces escribir 
+\begin_inset Formula $P=(p_{1},\dots,p_{n})$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v}=(v_{1},\dots,v_{n})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Para cambiar coordenadas entre dos referenciales 
+\begin_inset Formula $\Re=(O,{\cal B})$
+\end_inset
+
+ y 
+\begin_inset Formula $\Re'=(O',{\cal B}')$
+\end_inset
+
+ de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+, si llamamos 
+\begin_inset Formula $X_{0}:=[O]_{\Re'}=[\overrightarrow{O'O}]_{{\cal B}'}$
+\end_inset
+
+ y 
+\begin_inset Formula $M:=M_{{\cal B}'{\cal B}}$
+\end_inset
+
+, se tiene que:
+\begin_inset Formula 
+\[
+\left.\begin{array}{c}
+X=[P]_{\Re}=[\overrightarrow{OP}]_{{\cal B}}\\
+X'=[P]_{\Re'}=[\overrightarrow{O'P}]_{{\cal B}'}
+\end{array}\right\} \implies X'=[\overrightarrow{O'P}]_{{\cal B}'}=[\overrightarrow{O'O}]_{{\cal B}'}+[\overrightarrow{OP}]_{{\cal B}'}=X_{0}+M\cdot[\overrightarrow{OP}]_{{\cal B}}=X_{0}+MX
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $X=(x_{1},\dots,x_{n})$
+\end_inset
+
+, 
+\begin_inset Formula $X'=(x'_{1},\dots,x'_{n})$
+\end_inset
+
+, 
+\begin_inset Formula $X_{0}=(b_{1},\dots,b_{n})$
+\end_inset
+
+ y 
+\begin_inset Formula $M=(a_{ij})$
+\end_inset
+
+, llamamos 
+\series bold
+ecuaciones de cambio de coordenadas
+\series default
+ a las siguientes:
+\begin_inset Formula 
+\[
+\left\{ \begin{array}{ccc}
+x'_{1} & = & b_{1}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\
+ & \vdots\\
+x'_{n} & = & b_{n}+a_{n1}x_{1}+\dots+a_{nn}x_{n}
+\end{array}\right.
+\]
+
+\end_inset
+
+Podemos emplear la expresión matricial equivalente:
+\begin_inset Formula 
+\[
+\left(\begin{array}{c}
+1\\
+x'_{1}\\
+\vdots\\
+x'_{n}
+\end{array}\right)=\left(\begin{array}{cccc}
+1 & 0 & \cdots & 0\\
+b_{1} & a_{11} & \cdots & a_{1n}\\
+\vdots & \vdots & \ddots & \vdots\\
+b_{n} & a_{n1} & \cdots & a_{nn}
+\end{array}\right)\left(\begin{array}{c}
+1\\
+x_{1}\\
+\vdots\\
+x_{n}
+\end{array}\right)
+\]
+
+\end_inset
+
+O simplificadamente
+\begin_inset Formula 
+\[
+\left(\begin{array}{c}
+1\\
+\hline X'
+\end{array}\right)=\left(\begin{array}{c|c}
+1 & 0\\
+\hline X_{0} & M
+\end{array}\right)\left(\begin{array}{c}
+1\\
+\hline X
+\end{array}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Rectas y puntos alineados
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+recta
+\series default
+ que pasa por 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ con 
+\series bold
+dirección
+\series default
+ 
+\begin_inset Formula $<\vec{v}>$
+\end_inset
+
+, o 
+\series bold
+vector director
+\series default
+ 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+, es el conjunto 
+\begin_inset Formula $P+<\vec{v}>=\{P+\lambda\vec{v}\}_{\lambda\in K}$
+\end_inset
+
+.
+ Dos rectas 
+\begin_inset Formula $l$
+\end_inset
+
+ y 
+\begin_inset Formula $l'$
+\end_inset
+
+ son 
+\series bold
+paralelas
+\series default
+ (
+\begin_inset Formula $l\parallel l'$
+\end_inset
+
+) si sus vectores directores son proporcionales.
+ Propiedades: 
+\begin_inset Formula $\forall X\in{\cal E},l=P+<\vec{v}>$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $X\in l\iff\exists\lambda\in K:\overrightarrow{PX}=\lambda\vec{v}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall r\neq0,l=P+<r\vec{v}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall P'\in l,l=P'+<\vec{v}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall Q\in{\cal E},\exists!r:Q\in r\parallel l$
+\end_inset
+
+; 
+\begin_inset Formula $r:=Q+<\vec{v}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Recta que pasa por
+\series default
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+: 
+\begin_inset Formula $\forall A,B\in{\cal E},A\neq B,\exists!r:A,B\in r$
+\end_inset
+
+; 
+\begin_inset Formula $r:=AB:=A+<\overrightarrow{AB}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una serie de puntos de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ están 
+\series bold
+alineados
+\series default
+ si existe una recta que los contiene a todos.
+\end_layout
+
+\begin_layout Subsection
+Puntos medios y razón simple
+\end_layout
+
+\begin_layout Standard
+Si en 
+\begin_inset Formula $K$
+\end_inset
+
+ se tiene que 
+\begin_inset Formula $2=1+1\neq0$
+\end_inset
+
+, se define el 
+\series bold
+punto medio
+\series default
+ de 
+\begin_inset Formula $A,B\in{\cal E}$
+\end_inset
+
+ como
+\begin_inset Formula 
+\[
+\frac{A+B}{2}:=A+\frac{1}{2}\overrightarrow{AB}
+\]
+
+\end_inset
+
+Esto es simplemente una notación, pues no hemos definido suma ni producto
+ por escalares en 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+.
+ Propiedades: 
+\begin_inset Formula $\forall A,B\in{\cal E}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $M=\frac{A+B}{2}\iff\overrightarrow{AB}=2\overrightarrow{AM}\iff B=A+2\overrightarrow{AM}\iff\overrightarrow{MA}+\overrightarrow{MB}=\vec{0}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\overrightarrow{MA}+\overrightarrow{MB}=\overrightarrow{MA}+\overrightarrow{MA}+\overrightarrow{AB}=2\overrightarrow{MA}+\overrightarrow{AB}=2\overrightarrow{MA}+2\overrightarrow{AM}=2\overrightarrow{MM}=\vec{0}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{A+B}{2}=\frac{B+A}{2}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\frac{A+B}{2}=A+\frac{1}{2}\overrightarrow{AB}=B+\overrightarrow{BA}-\frac{1}{2}\overrightarrow{BA}=B+\frac{1}{2}\overrightarrow{BA}=\frac{B+A}{2}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{A+B}{2}=\frac{A+B'}{2}\iff B=B'$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+A+\frac{1}{2}\overrightarrow{AB}=A+\frac{1}{2}\overrightarrow{AB'}\iff\overrightarrow{AB}=\overrightarrow{AB'}\iff B=B'
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\frac{(A+\vec{v})+(B+\vec{w})}{2}=\frac{A+B}{2}+\frac{\vec{v}+\vec{w}}{2}$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A+\vec{v}+\frac{1}{2}\overrightarrow{(A+\vec{v})(B+\vec{w})}=A+\vec{v}+\frac{1}{2}(\overrightarrow{AB}+\vec{w}-\vec{v})=\left(A+\frac{1}{2}\overrightarrow{AB}\right)+\frac{1}{2}(\vec{v}+\vec{w})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dados tres puntos alineados 
+\begin_inset Formula $A,B,C$
+\end_inset
+
+ con 
+\begin_inset Formula $A\neq B$
+\end_inset
+
+ y 
+\begin_inset Formula $C\in AB$
+\end_inset
+
+, llamamos 
+\series bold
+razón simple
+\series default
+ de 
+\begin_inset Formula $A,B,C$
+\end_inset
+
+ al único 
+\begin_inset Formula $\lambda\in K$
+\end_inset
+
+ con 
+\begin_inset Formula $\overrightarrow{AC}=\lambda\overrightarrow{AB}$
+\end_inset
+
+, y escribimos 
+\begin_inset Formula $\lambda=(A,B,C)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $(A,B,A)=0$
+\end_inset
+
+ y 
+\begin_inset Formula $(A,B,B)=1$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/gae/n2.lyx b/gae/n2.lyx
new file mode 100644
index 0000000..d5c7289
--- /dev/null
+++ b/gae/n2.lyx
@@ -0,0 +1,1978 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Una aplicación 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
+\end_inset
+
+ es 
+\series bold
+afín
+\series default
+ si existe 
+\begin_inset Formula $\overrightarrow{f}:V\rightarrow V'$
+\end_inset
+
+ tal que para 
+\begin_inset Formula $P\in{\cal E},\vec{v}\in V$
+\end_inset
+
+, 
+\begin_inset Formula $f(P+\vec{v})=f(P)+\overrightarrow{f}(\vec{v})$
+\end_inset
+
+, es decir, 
+\begin_inset Formula $\overrightarrow{f}(\vec{v})=\overrightarrow{f(P)f(P+\vec{v})}$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ queda determinada por 
+\begin_inset Formula $f$
+\end_inset
+
+ y se le llama 
+\series bold
+aplicación lineal asociada
+\series default
+ a 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Las aplicaciones afines 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ son 
+\series bold
+transformaciones afines
+\series default
+ de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal E}'$
+\end_inset
+
+ tienen dimensión finita siendo 
+\begin_inset Formula $\Re=(O;{\cal B}=\{\vec{v}_{1},\dots,\vec{v}_{n}\})$
+\end_inset
+
+ y 
+\begin_inset Formula $\Re'=(O';{\cal B}')$
+\end_inset
+
+ referenciales cartesianos de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal E}'$
+\end_inset
+
+, sea 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
+\end_inset
+
+ con 
+\begin_inset Formula $X_{0}=[f(O)]_{\Re'}=[\overrightarrow{O'f(O)}]_{{\cal B}'}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $M=M_{{\cal B}'{\cal B}}(\overrightarrow{f})$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+[f(X)]_{\Re'}=[f(O)+\overrightarrow{f}(\overrightarrow{OX})]_{\Re'}=[f(O)]_{\Re'}+[\overrightarrow{f}(\overrightarrow{OX})]_{{\cal B}}=X_{0}+M[X]_{\Re}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lo que nos da la 
+\series bold
+representación matricial
+\series default
+ o las 
+\series bold
+ecuaciones
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $\Re$
+\end_inset
+
+ y 
+\begin_inset Formula $\Re'$
+\end_inset
+
+ como 
+\begin_inset Formula $X'=X_{0}+MX$
+\end_inset
+
+ o
+\begin_inset Formula 
+\[
+\left(\begin{array}{c}
+1\\
+\hline X'
+\end{array}\right)=\left(\begin{array}{c|c}
+1 & 0\\
+\hline X_{0} & M
+\end{array}\right)\left(\begin{array}{c}
+1\\
+\hline X
+\end{array}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Propiedades
+\end_layout
+
+\begin_layout Enumerate
+Dados 
+\begin_inset Formula $f,g:{\cal E}\rightarrow{\cal E}'$
+\end_inset
+
+, 
+\begin_inset Formula $\exists P\in{\cal E}:f(P)=g(P)\land\overrightarrow{f}=\overrightarrow{g}\implies f=g$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Dado un 
+\begin_inset Formula $Q\in{\cal E}$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $f(Q)=f(P)+\overrightarrow{f}(\overrightarrow{PQ})=g(P)+\overrightarrow{g}(\overrightarrow{PQ})=g(Q)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dados 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+, 
+\begin_inset Formula $P'\in{\cal E}'$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi:V\rightarrow V'$
+\end_inset
+
+ vectorial, existe una única aplicación afín 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
+\end_inset
+
+ con 
+\begin_inset Formula $f(P)=P'$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{f}=\phi$
+\end_inset
+
+, dada por 
+\begin_inset Formula $f(Q):=P'+\phi(\overrightarrow{PQ})$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+f(Q+\vec{v})=P'+\phi(\overrightarrow{P(Q+\vec{v})})=P'+\phi(\overrightarrow{PQ}+\vec{v})=P'+\phi(\overrightarrow{PQ})+\phi(\vec{v})=f(Q)+\phi(\vec{v})
+\]
+
+\end_inset
+
+por lo que es afín.
+ Además, 
+\begin_inset Formula $f(P)=P'+\phi(\overrightarrow{PP})=P'$
+\end_inset
+
+, y la unicidad se desprende del apartado anterior.
+\end_layout
+
+\begin_layout Enumerate
+La composición de aplicaciones afines 
+\begin_inset Formula $g$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es afín, y 
+\begin_inset Formula $\overrightarrow{g\circ f}=\overrightarrow{g}\circ\overrightarrow{f}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula ${\cal E}\overset{f}{\rightarrow}{\cal E}'\overset{g}{\rightarrow}{\cal E}''$
+\end_inset
+
+, para 
+\begin_inset Formula $P\in{\cal E},\vec{v}\in V$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+(g\circ f)(P+\vec{v})=g(f(P)+\overrightarrow{f}(\vec{v}))=g(f(P))+\overrightarrow{g}(\overrightarrow{f}(\vec{v}))=(g\circ f)(P)+(\overrightarrow{g}\circ\overrightarrow{f})(\vec{v})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectiva si y sólo si lo es 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dados 
+\begin_inset Formula $P\in{\cal E},\vec{v}\in\text{Nuc}(\overrightarrow{f})$
+\end_inset
+
+, 
+\begin_inset Formula $f(P+\vec{v})=f(P)+\overrightarrow{f}(\vec{v})=f(P)$
+\end_inset
+
+, y por la inyectividad 
+\begin_inset Formula $P+\vec{v}=P$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v}=0$
+\end_inset
+
+, de modo que 
+\begin_inset Formula $\text{Nuc}(\overrightarrow{f})=\{0\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $f(P)=f(Q)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\overrightarrow{f}(\overrightarrow{PQ})=\overrightarrow{f(P)f(Q)}=\vec{0}$
+\end_inset
+
+, y por la inyectividad de 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+, 
+\begin_inset Formula $\overrightarrow{PQ}=\vec{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $P=Q$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $f$
+\end_inset
+
+ es suprayectiva si y sólo si lo es 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dado 
+\begin_inset Formula $\vec{v}'\in V'$
+\end_inset
+
+, sea 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $f(P)+\vec{v}'\in{\cal E}'$
+\end_inset
+
+ y por la suprayectividad de 
+\begin_inset Formula $f$
+\end_inset
+
+, existe 
+\begin_inset Formula $Q\in{\cal E}$
+\end_inset
+
+ con 
+\begin_inset Formula $f(Q)=f(P)+\vec{v}'$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\vec{v}'=\overrightarrow{f(P)f(Q)}=\overrightarrow{f}(\overrightarrow{PQ})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dado 
+\begin_inset Formula $Q'\in{\cal E}'$
+\end_inset
+
+, sea 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $\overrightarrow{f(P)Q'}\in V'$
+\end_inset
+
+, y por la suprayectividad de 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ existe 
+\begin_inset Formula $\vec{v}\in V$
+\end_inset
+
+ con 
+\begin_inset Formula $\overrightarrow{f}(\vec{v})=\overrightarrow{f(P)Q'}$
+\end_inset
+
+, luego 
+\begin_inset Formula $Q'=f(P)+\overrightarrow{f}(\vec{v})=f(P+\vec{v})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
+\end_inset
+
+ es afín y biyectiva, entonces 
+\begin_inset Formula $f^{-1}$
+\end_inset
+
+ es afín y 
+\begin_inset Formula $\overrightarrow{f^{-1}}=\overrightarrow{f}^{-1}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula 
+\[
+f^{-1}(P'+\vec{v}')=f^{-1}(P')+\overrightarrow{f}^{-1}(\vec{v}')\iff f(f^{-1}(P'+\vec{v}'))=P'+\vec{v}'=f(f^{-1}(P')+\overrightarrow{f}^{-1}(\vec{v}'))
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Esto último nos lleva al concepto de 
+\series bold
+isomorfismo de espacios afines
+\series default
+, una aplicación afín y biyectiva 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}'$
+\end_inset
+
+.
+ Cuando existe se dice que 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal E}'$
+\end_inset
+
+ son 
+\series bold
+isomorfos
+\series default
+.
+ Como 
+\series bold
+teorema
+\series default
+, dos espacios afines de dimensión finita sobre el mismo cuerpo son isomorfos
+ si y sólo si tienen la misma dimensión.
+ Más propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $M=\frac{A+B}{2}\implies f(M)=\frac{f(A)+f(B)}{2}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\overrightarrow{AB}=2\overrightarrow{AM}\implies\overrightarrow{f(A)f(B)}=\overrightarrow{f}(\overrightarrow{AB})=\overrightarrow{f}(2\overrightarrow{AM})=2\overrightarrow{f(A)f(M)}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula ${\cal L}=P+W$
+\end_inset
+
+ es una variedad de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+, 
+\begin_inset Formula $f({\cal L})=f(P)+\overrightarrow{f}(W)$
+\end_inset
+
+ lo es de 
+\begin_inset Formula ${\cal E}'$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\begin{array}{c}
+Q'\in f({\cal L})\iff\exists\vec{w}\in W:Q'=f(P+\vec{w})=f(P)+\overrightarrow{f}(\vec{w})\iff\\
+\iff\overrightarrow{f(P)Q'}=\overrightarrow{f}(\vec{w})\in\overrightarrow{f}(W)\iff Q'\in f(P)+\overrightarrow{f}(W)
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}_{1}\ll{\cal L}_{2}\subseteq{\cal E}\implies f({\cal L}_{1})\ll f({\cal L}_{2})$
+\end_inset
+
+; 
+\begin_inset Formula ${\cal L}_{1}\parallel{\cal L}_{2}\subseteq{\cal E}\implies f({\cal L}_{1})\parallel f({\cal L}_{2})$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Se sigue de lo anterior y de que 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ conserva las inclusiones entre subespacios.
+\end_layout
+
+\begin_layout Enumerate
+Sea 
+\begin_inset Formula $f$
+\end_inset
+
+ biyectiva, si 
+\begin_inset Formula ${\cal L}'=P'+W$
+\end_inset
+
+ es una variedad de 
+\begin_inset Formula ${\cal E}'$
+\end_inset
+
+ y su inversa 
+\begin_inset Formula $f^{-1}({\cal L}')\neq\emptyset$
+\end_inset
+
+, esta es una variedad de 
+\begin_inset Formula ${\cal E}$
+\end_inset
+
+.
+ En concreto, 
+\begin_inset Formula $\text{dir}(f^{-1}({\cal L}'))=\overrightarrow{f}^{-1}(W')$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\begin{array}{c}
+Q\in f^{-1}({\cal L}')\iff f(Q)\in{\cal L}'\iff\overrightarrow{P'f(Q)}\in W'\iff\\
+\iff\overrightarrow{f(P)P'}+\overrightarrow{P'f(Q)}=\overrightarrow{f(P)f(Q)}=\overrightarrow{f}(\overrightarrow{PQ})\in W'\iff\\
+\iff\overrightarrow{PQ}\in\overrightarrow{f}^{-1}(W')\iff Q\in P+\overrightarrow{f}^{-1}(W')
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Puntos fijos
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Q\in{\cal E}$
+\end_inset
+
+ es un 
+\series bold
+punto fijo
+\series default
+ de 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ si 
+\begin_inset Formula $f(Q)=Q$
+\end_inset
+
+, y definimos
+\begin_inset Formula 
+\[
+\text{Fix}(f):=\{Q\in{\cal E}:f(Q)=Q\}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+subespacio invariante
+\series default
+ por 
+\begin_inset Formula $\phi:V\rightarrow V$
+\end_inset
+
+ es un subespacio 
+\begin_inset Formula $U$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ con 
+\begin_inset Formula $f(U)\subseteq U$
+\end_inset
+
+.
+ Destacamos el subespacio de los 
+\series bold
+vectores invariantes
+\series default
+ o asociado al autovalor 1,
+\begin_inset Formula 
+\[
+\text{Inv}(\phi):=\text{Nuc}(\phi-id_{V})=\{\vec{v}\in V:\phi(\vec{v})=\vec{v}\}
+\]
+
+\end_inset
+
+y el de los 
+\series bold
+opuestos
+\series default
+ o asociado al autovalor 
+\begin_inset Formula $-1$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\text{Opp}(\phi):=\text{Nuc}(\phi+id_{V})=\{\vec{v}\in V:\phi(\vec{v})=-\vec{v}\}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Se tiene que 
+\begin_inset Formula $P\in\text{Fix}(f)\neq\emptyset\implies\text{Fix}(f)=P+\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $f(P)=P$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\begin{array}{c}
+Q\in P+\text{Inv}(\overrightarrow{f})\iff\overrightarrow{PQ}\in\text{Inv}(\overrightarrow{f})\iff\overrightarrow{PQ}=\overrightarrow{f}(\overrightarrow{PQ})=\overrightarrow{f(P)f(Q)}=\overrightarrow{Pf(Q)}\iff\\
+\iff Q=f(Q)\iff Q\in\text{Fix}(f)
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En coordenadas, 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+ se obtiene como las soluciones del sistema 
+\begin_inset Formula $(I-M|0)$
+\end_inset
+
+, mientras que 
+\begin_inset Formula $\text{Fix}(f)$
+\end_inset
+
+ se obtiene como las soluciones del sistema 
+\begin_inset Formula $(I-M|X_{0})$
+\end_inset
+
+.
+ Por tanto, 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})=0\iff|\text{Fix}(f)|=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Ejemplos de transformaciones afines
+\end_layout
+
+\begin_layout Subsection
+Traslaciones 
+\end_layout
+
+\begin_layout Standard
+Dado 
+\begin_inset Formula $\vec{v}\in V$
+\end_inset
+
+, la 
+\series bold
+traslación
+\series default
+ de vector 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ es la aplicación 
+\begin_inset Formula $t_{\vec{v}}:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ con 
+\begin_inset Formula $t_{\vec{v}}(P)=P+\vec{v}$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $t_{\vec{v}}$
+\end_inset
+
+ es afín y 
+\begin_inset Formula $\overrightarrow{t_{\vec{v}}}=id_{V}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+t_{\vec{v}}(P+\vec{w})=P+\vec{w}+\vec{v}=t_{\vec{v}}(P)+id_{V}(\vec{w})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Recíprocamente, si 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ es afín con 
+\begin_inset Formula $\overrightarrow{f}=id_{V}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f=t_{\overrightarrow{Pf(P)}}$
+\end_inset
+
+, dado 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sea 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ arbitrario y 
+\begin_inset Formula $\vec{v}:=\overrightarrow{Pf(P)}$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $t_{\vec{v}}$
+\end_inset
+
+ son aplicaciones afines con la misma lineal asociada y actúan igual sobre
+ 
+\begin_inset Formula $P$
+\end_inset
+
+, luego 
+\begin_inset Formula $f=t_{\vec{v}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $t_{\vec{0}}=id_{{\cal E}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\vec{v}\neq\vec{0}\implies\text{Fix}(t_{\vec{v}})=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $t_{\vec{v}}\circ t_{\vec{w}}=t_{\vec{w}}\circ t_{\vec{v}}=t_{\vec{v}+\vec{w}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $t_{\vec{v}}^{-1}=t_{-\vec{v}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La expresión matricial de 
+\begin_inset Formula $t_{\vec{v}}$
+\end_inset
+
+ sobre 
+\begin_inset Formula $\Re=(O,{\cal B})$
+\end_inset
+
+ es 
+\begin_inset Formula $X'=[\vec{v}]_{{\cal B}}+X$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ afín, 
+\begin_inset Formula $f\circ t_{\vec{v}}=t_{\vec{v}}\circ f\iff\vec{v}\in\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Como ambas tienen la misma lineal asociada (
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+), serán iguales si y sólo si actúan igual sobre un 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ arbitrario.
+\begin_inset Formula 
+\[
+f\circ t_{\vec{v}}=t_{\vec{v}}\circ f\iff f(t_{\vec{v}}(P))=t_{\vec{v}}(f(P))\iff f(P+\vec{v})=f(P)+\vec{v}\iff\overrightarrow{f}(\vec{v})=\vec{v}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Dado 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ y 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+, 
+\begin_inset Formula $f=t_{\vec{v}}\circ g$
+\end_inset
+
+ donde 
+\begin_inset Formula $\vec{v}=\overrightarrow{Pf(P)}$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ es una transformación afín con 
+\begin_inset Formula $g(P)=P$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{g}=\overrightarrow{f}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $g:=t_{-\vec{v}}\circ f$
+\end_inset
+
+ es afín con 
+\begin_inset Formula $g(P)=t_{-\vec{v}}(f(P))=f(P)-\vec{v}=f(P)+\overrightarrow{f(P)P}=P$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{g}=\overrightarrow{t_{-\vec{v}}}\circ\overrightarrow{f}=\overrightarrow{f}$
+\end_inset
+
+, y componiendo se obtiene 
+\begin_inset Formula $f=t_{\vec{v}}\circ g$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Homotecias
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula $O\in{\cal E},\lambda\in K$
+\end_inset
+
+, la 
+\series bold
+homotecia
+\series default
+ de centro 
+\begin_inset Formula $O$
+\end_inset
+
+ y razón 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ es la aplicación 
+\begin_inset Formula $H_{O,\lambda}:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $H_{O,\lambda}(P):=O+\lambda\overrightarrow{OP}$
+\end_inset
+
+.
+ Así, para 
+\begin_inset Formula $P\neq O$
+\end_inset
+
+, la razón simple 
+\begin_inset Formula $(O,P,H_{O,\lambda}(P))=\lambda$
+\end_inset
+
+.
+ Para 
+\begin_inset Formula $\lambda=0$
+\end_inset
+
+ se obtiene la aplicación constante, que lleva todos los puntos a 
+\begin_inset Formula $O$
+\end_inset
+
+; para 
+\begin_inset Formula $\lambda=1$
+\end_inset
+
+ se obtiene la identidad, y para 
+\begin_inset Formula $\lambda=-1$
+\end_inset
+
+ se obtiene la 
+\series bold
+simetría central
+\series default
+ sobre 
+\begin_inset Formula $O$
+\end_inset
+
+, escrita 
+\begin_inset Formula $s_{O}:=H_{O,-1}$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $H_{O,\lambda}$
+\end_inset
+
+ es afín con 
+\begin_inset Formula $\overrightarrow{H_{O,\lambda}}=h_{\lambda}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+H_{O,\lambda}(P+\vec{w})=O+\lambda\overrightarrow{O(P+\vec{w})}=O+\lambda(\overrightarrow{OP}+\overrightarrow{w})=(O+\lambda\overrightarrow{OP})+\lambda\vec{w}=H_{O,\lambda}(P)+h_{\lambda}(\vec{w})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\lambda\neq1\implies\text{Fix}(H_{O,\lambda})=\{O\}$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\begin{array}{c}
+P=H_{O,\lambda}(P)=O+\lambda\overrightarrow{OP}\iff\overrightarrow{OP}=\lambda\overrightarrow{OP}\iff\\
+\iff(\lambda-1)\overrightarrow{OP}=\vec{0}\overset{\lambda\neq1}{\iff}\overrightarrow{OP}=\vec{0}\iff P=O
+\end{array}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ es afín con 
+\begin_inset Formula $\overrightarrow{f}=h_{\lambda}$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\neq1$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es la homotecia 
+\begin_inset Formula $f=H_{O,\lambda}$
+\end_inset
+
+ con 
+\begin_inset Formula $O=P+\frac{1}{1-\lambda}\overrightarrow{Pf(P)}$
+\end_inset
+
+.
+ Así, para una simetría central, 
+\begin_inset Formula $O=\frac{P+f(P)}{2}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Como 
+\begin_inset Formula $\overrightarrow{f}=\overrightarrow{H_{O,\lambda}}$
+\end_inset
+
+, será 
+\begin_inset Formula $f=H_{O,\lambda}$
+\end_inset
+
+ si actúan igual sobre un punto.
+ Por la definición de 
+\begin_inset Formula $O$
+\end_inset
+
+ se tiene que 
+\begin_inset Formula $\overrightarrow{PO}=\frac{1}{1-\lambda}\overrightarrow{Pf(P)}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $(1-\lambda)\overrightarrow{PO}=\overrightarrow{Pf(P)}$
+\end_inset
+
+, luego
+\begin_inset Formula 
+\[
+\overrightarrow{Of(O)}=\overrightarrow{OP}+\overrightarrow{Pf(P)}+\overrightarrow{f(P)f(O)}=-\overrightarrow{PO}+(1-\lambda)\overrightarrow{PO}+\lambda\overrightarrow{PO}=\vec{0}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $H_{O,\lambda}\circ H_{O,\mu}=H_{O,\mu}\circ H_{O,\lambda}=H_{O,\lambda\mu}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\lambda\neq0\implies H_{O,\lambda}^{-1}=H_{O,\lambda^{-1}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La expresión matricial de 
+\begin_inset Formula $H_{O,\lambda}$
+\end_inset
+
+ en el referencial 
+\begin_inset Formula $\Re$
+\end_inset
+
+ es 
+\begin_inset Formula $X'=(1-\lambda)[O]_{\Re}+\lambda X$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\lambda\neq1$
+\end_inset
+
+ entonces 
+\begin_inset Formula $t_{\vec{v}}\circ H_{O,\lambda}$
+\end_inset
+
+ y 
+\begin_inset Formula $H_{O,\lambda}\circ t_{\vec{v}}$
+\end_inset
+
+ son homotecias de razón 
+\begin_inset Formula $\lambda$
+\end_inset
+
+ y centros respectivos 
+\begin_inset Formula $O+\frac{1}{1-\lambda}\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $O+\frac{\lambda}{1-\lambda}\vec{v}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $O\neq O'$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\lambda'=1$
+\end_inset
+
+ entonces 
+\begin_inset Formula $H_{O,\lambda}\circ H_{O',\lambda'}=t_{(1-\lambda)\overrightarrow{O'O}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Proyecciones y simetrías vectoriales
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $V=W_{1}\oplus W_{2}$
+\end_inset
+
+, la 
+\series bold
+proyección vectorial
+\series default
+ 
+\begin_inset Formula $\pi$
+\end_inset
+
+ y la 
+\series bold
+simetría vectorial
+\series default
+ 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ de 
+\series bold
+base
+\series default
+ 
+\begin_inset Formula $W_{1}$
+\end_inset
+
+ y 
+\series bold
+dirección
+\series default
+ 
+\begin_inset Formula $W_{2}$
+\end_inset
+
+, o sobre 
+\begin_inset Formula $W_{1}$
+\end_inset
+
+ y paralelamente a 
+\begin_inset Formula $W_{2}$
+\end_inset
+
+ son los endomorfismos de 
+\begin_inset Formula $V$
+\end_inset
+
+ tales que, si 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ se descompone como 
+\begin_inset Formula $\vec{v}=\vec{w}_{1}+\vec{w}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{w}_{1}\in W_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}_{2}\in W_{2}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\pi_{W_{1},W_{2}}(\vec{v})=\vec{w}_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma_{W_{1},W_{2}}(\vec{v})=\vec{w}_{1}-\vec{w}_{2}$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\sigma+id_{V}=2\pi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\pi$
+\end_inset
+
+ es 
+\series bold
+idempotente
+\series default
+ (
+\begin_inset Formula $\pi^{2}=\pi$
+\end_inset
+
+) y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ es 
+\series bold
+involutiva
+\series default
+ (
+\begin_inset Formula $\sigma^{2}=id_{V}$
+\end_inset
+
+, es decir, 
+\begin_inset Formula $\sigma^{-1}=\sigma$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $W_{1}=\text{Inv}(\pi)$
+\end_inset
+
+ y 
+\begin_inset Formula $W_{2}=\text{Nuc}(\pi)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $W_{1}=\text{Inv}(\sigma)$
+\end_inset
+
+ y 
+\begin_inset Formula $W_{2}=\text{Opp}(\sigma)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\phi\text{ es proyección (con }W_{1}=\text{Inv}(\phi)\text{ y }W_{2}=\text{Nuc}(\phi)\text{)}\iff\phi\text{ es \textbf{idempotente} (}\phi^{2}=\phi\text{)}\iff V=\text{Inv}(\phi)\oplus\text{Nuc}(\phi)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[2\implies3]$
+\end_inset
+
+ 
+\begin_inset Formula $\vec{v}=\phi(\vec{v})+(\vec{v}-\phi(\vec{v}))\in\text{Inv}(\phi)+\text{Nuc}(\phi)$
+\end_inset
+
+ para todo 
+\begin_inset Formula $\vec{v}\in V$
+\end_inset
+
+, y 
+\begin_inset Formula $\vec{v}\in\text{Inv}(\phi)\cap\text{Nuc}(\phi)\implies\vec{v}\overset{\text{Inv}}{=}\phi(\vec{v})\overset{\text{Nuc}}{=}\vec{0}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[3\implies1]$
+\end_inset
+
+ Si 
+\begin_inset Formula $\vec{v}=\vec{w}_{1}+\vec{w}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{w}_{1}\in\text{Inv}(\phi)$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}_{2}\in\text{Nuc}(\phi)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\phi(\vec{v})=\phi(\vec{w}_{1})+\phi(\vec{w}_{2})=\vec{w}_{1}+\vec{0}=\vec{w}_{1}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\phi$
+\end_inset
+
+ es la proyección de base 
+\begin_inset Formula $\text{Inv}(\phi)$
+\end_inset
+
+ y dirección 
+\begin_inset Formula $\text{Nuc}(\phi)$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\phi\text{ es simetría (con }W_{1}=\text{Inv}(\phi)\text{ y }W_{2}=\text{Nuc}(\phi)\text{)}\iff\text{\phi}\text{ es \textbf{involutiva} (}\phi^{2}=id_{V}\text{)}\iff V=\text{Inv}(\phi)\oplus\text{Opp}(\phi)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Demostración análoga, tomando 
+\begin_inset Formula $\vec{v}=\frac{1}{2}(\vec{v}+\phi(\vec{v}))+\frac{1}{2}(\vec{v}-\phi(\vec{v}))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\{\vec{w}_{1},\dots,\vec{w}_{n}\}$
+\end_inset
+
+ es base de 
+\begin_inset Formula $W_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{\vec{u}_{1},\dots,\vec{u}_{m}\}$
+\end_inset
+
+ es de 
+\begin_inset Formula $W_{2}$
+\end_inset
+
+, podemos definir la base 
+\begin_inset Formula ${\cal B}:=\{\vec{w}_{1},\dots,\vec{w}_{n},\vec{u}_{1},\dots,\vec{u}_{m}\}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $M_{{\cal B}}(\pi_{W_{1},W_{2}})=\left(\begin{array}{c|c}
+I_{n} & 0\\
+\hline 0 & 0
+\end{array}\right)$
+\end_inset
+
+ y 
+\begin_inset Formula $M_{{\cal B}}(\sigma_{W_{1},W_{2}})=\left(\begin{array}{c|c}
+I_{n} & 0\\
+\hline 0 & -I_{m}
+\end{array}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Proyecciones y simetrías afines
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula ${\cal L}=A+W_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $V=W_{1}\oplus W_{2}$
+\end_inset
+
+, la 
+\series bold
+proyección afín
+\series default
+ 
+\begin_inset Formula $p$
+\end_inset
+
+ y la 
+\series bold
+simetría afín
+\series default
+ 
+\begin_inset Formula $s$
+\end_inset
+
+ sobre 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ paralelamente a 
+\begin_inset Formula $W_{2}$
+\end_inset
+
+ son las aplicaciones 
+\begin_inset Formula $p_{{\cal L},W_{2}},s_{{\cal L},W_{2}}:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+ tales que 
+\begin_inset Formula $p(Q)\in{\cal L}\cap(Q+W_{2})$
+\end_inset
+
+ (conjunto unitario porque las variedades son complementarias) y 
+\begin_inset Formula $s(Q)=p(Q)+\overrightarrow{Qp(Q)}=Q+2\overrightarrow{Qp(Q)}$
+\end_inset
+
+.
+ Visto de otro modo, si 
+\begin_inset Formula $Q=A+\vec{w}_{1}+\vec{w}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{w}_{1}\in W_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}_{2}\in W_{2}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $p(Q)=A+\vec{w}_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $s(Q)=A+\vec{w}_{1}-\vec{w}_{2}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula ${\cal L}=\{O\}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $p$
+\end_inset
+
+ es la aplicación constante en 
+\begin_inset Formula $O$
+\end_inset
+
+ y 
+\begin_inset Formula $s$
+\end_inset
+
+ es la simetría central de centro 
+\begin_inset Formula $O$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $p_{{\cal L},W_{2}}$
+\end_inset
+
+ y 
+\begin_inset Formula $s_{{\cal L},W_{2}}$
+\end_inset
+
+ son afines con 
+\begin_inset Formula $\overrightarrow{p_{{\cal L},W_{2}}}=\pi_{W_{1},W_{2}}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{s_{{\cal L},W_{2}}}=\sigma_{W_{1},W_{2}}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula $\overrightarrow{AQ}=\vec{w}_{1}+\vec{w}_{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{u}=\vec{u}_{1}+\vec{u}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{u}_{1},\vec{w}_{1}\in W_{1},\vec{u}_{2},\vec{w}_{2}\in W_{2}$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+p(Q+\vec{u})=p(A+(\vec{w}_{1}+\vec{u}_{1})+(\vec{w}_{2}+\vec{u}_{2}))=A+(\vec{w}_{1}+\vec{u}_{1})=(A+\vec{w}_{1})+\vec{u}_{1}=p(A)+\pi(\vec{u})
+\]
+
+\end_inset
+
+La simetría se hace de forma análoga.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}=\text{Fix}(p)$
+\end_inset
+
+ y 
+\begin_inset Formula $W_{2}=\text{Nuc}(\pi)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Si 
+\begin_inset Formula $\vec{w}_{1}\in W_{1},\vec{w}_{2}\in W_{2}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+Q:=A+\vec{w}_{1}+\vec{w}_{2}\in\text{Fix}(p)\iff\vec{w}_{2}=0\iff Q=A+\vec{w}_{1}\iff Q\in{\cal L}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}=\text{Fix}(s)$
+\end_inset
+
+ y 
+\begin_inset Formula $W_{2}=\text{Opp}(\sigma)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dada una transformación afín 
+\begin_inset Formula $f:{\cal E}\rightarrow{\cal E}$
+\end_inset
+
+, 
+\begin_inset Formula $f\text{ es una proyección afín (con }{\cal L}=\text{Fix}(f)\text{ y }W_{2}=\text{Nuc}(\overrightarrow{f})\text{)}\iff f\text{ es idempotente}\iff\overrightarrow{f}^{2}=\overrightarrow{f}\land\text{Fix}(f)\neq\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[1\implies2]$
+\end_inset
+
+ 
+\begin_inset Formula $f^{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ actúan igual sobre los puntos de 
+\begin_inset Formula $\text{Fix}(f)\neq\emptyset$
+\end_inset
+
+, pues ambas los fijan, y 
+\begin_inset Formula $\overrightarrow{f^{2}}=\overrightarrow{f}^{2}=\overrightarrow{f}$
+\end_inset
+
+, luego 
+\begin_inset Formula $f^{2}=f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[2\implies3]$
+\end_inset
+
+ 
+\begin_inset Formula $\overrightarrow{f}^{2}=\overrightarrow{f^{2}}\overset{f^{2}=f}{=}\overrightarrow{f}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ es proyección vectorial.
+ Por otro lado, dado 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+, 
+\begin_inset Formula $f(P)=f(f(P))\in\text{Fix}(f)\neq\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[3\implies1]$
+\end_inset
+
+ Sea 
+\begin_inset Formula $A\in\text{Fix}(f)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{Fix}(f)=A+\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+, pero 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ es la proyección de base 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+ y dirección 
+\begin_inset Formula $\text{Nuc}(\overrightarrow{f})$
+\end_inset
+
+.
+ Ahora bien, dados 
+\begin_inset Formula $\vec{w}_{1}\in\text{Inv}(\overrightarrow{f}),\vec{w}_{2}\in\text{Nuc}(\overrightarrow{f})$
+\end_inset
+
+, 
+\begin_inset Formula $f(A+\vec{w}_{1}+\vec{w}_{2})=f(A)+\overrightarrow{f}(\vec{w}_{1}+\vec{w}_{2})=A+\vec{w}_{1}$
+\end_inset
+
+, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ es la proyección de base 
+\begin_inset Formula $A+\text{Inv}(\overrightarrow{f})=\text{Fix}(f)$
+\end_inset
+
+ y dirección 
+\begin_inset Formula $\text{Nuc}(\overrightarrow{f})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dada una transformación afín 
+\begin_inset Formula $f$
+\end_inset
+
+, 
+\begin_inset Formula $f\text{ es una simetría afín (con }{\cal L}=\text{Fix}(f)\text{ y }W_{2}=\text{Opp}(\overrightarrow{f})\text{)}\iff f\text{ es involutiva}\iff\overrightarrow{f}^{2}=id_{V}\land\text{Fix}(f)\neq\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[1\implies2]$
+\end_inset
+
+ 
+\begin_inset Formula $f^{2}$
+\end_inset
+
+ e 
+\begin_inset Formula $id_{{\cal E}}$
+\end_inset
+
+ actúan igual sobre los puntos de 
+\begin_inset Formula $\text{Fix}(f)$
+\end_inset
+
+, pues ambos los fijan, y 
+\begin_inset Formula $\overrightarrow{f^{2}}=\overrightarrow{f}^{2}=id_{V}$
+\end_inset
+
+, luego 
+\begin_inset Formula $f^{2}=f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[2\implies3]$
+\end_inset
+
+ 
+\begin_inset Formula $\overrightarrow{f}^{2}=\overrightarrow{f^{2}}=\overrightarrow{id_{{\cal E}}}=id_{V}$
+\end_inset
+
+.
+ Por otro lado, dado 
+\begin_inset Formula $P\in{\cal E}$
+\end_inset
+
+ y sea 
+\begin_inset Formula $A:=\frac{P+f(P)}{2}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f(A)=\frac{f(P)+f(f(P))}{2}=\frac{f(P)+P}{2}=A\in\text{Fix}(f)\neq\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $[3\implies1]$
+\end_inset
+
+ Sea 
+\begin_inset Formula $A\in\text{Fix}(f)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{Fix}(f)=A+\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+, pero 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ es la simetría de base 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+ y dirección 
+\begin_inset Formula $\text{Opp}(\overrightarrow{f})$
+\end_inset
+
+.
+ Ahora bien, dados 
+\begin_inset Formula $\vec{w}_{1}\in\text{Inv}(\overrightarrow{f}),\vec{w}_{2}\in\text{Opp}(\overrightarrow{f})$
+\end_inset
+
+, 
+\begin_inset Formula $f(A+\vec{w}_{1}+\vec{w}_{2})=f(A)+\overrightarrow{f}(\vec{w}_{1}+\vec{w}_{2})=A+\vec{w}_{1}-\vec{w}_{2}$
+\end_inset
+
+, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ es la simetría de base 
+\begin_inset Formula $A+\text{Inv}(\overrightarrow{f})=\text{Fix}(f)$
+\end_inset
+
+ y dirección 
+\begin_inset Formula $\text{Opp}(\overrightarrow{f})$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/gae/n3.lyx b/gae/n3.lyx
new file mode 100644
index 0000000..142ff61
--- /dev/null
+++ b/gae/n3.lyx
@@ -0,0 +1,1832 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
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+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
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+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Un 
+\series bold
+producto escalar
+\series default
+ en un espacio vectorial 
+\begin_inset Formula $V$
+\end_inset
+
+ es una aplicación 
+\begin_inset Formula $V\times V\rightarrow\mathbb{R}$
+\end_inset
+
+, representada por 
+\begin_inset Formula $(\vec{v},\vec{w})\mapsto\vec{v}\cdot\vec{w}$
+\end_inset
+
+, que verifica que 
+\begin_inset Formula $\forall\vec{u},\vec{v},\vec{w}\in V$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Es 
+\series bold
+simétrico
+\series default
+: 
+\begin_inset Formula $\vec{u}\cdot\vec{v}=\vec{v}\cdot\vec{u}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Es 
+\series bold
+lineal
+\series default
+ (en cada variable): 
+\begin_inset Formula $\vec{u}\cdot(\vec{v}+\lambda\vec{w})=\vec{u}\cdot\vec{v}+\lambda\vec{u}\cdot\vec{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Es 
+\series bold
+definido positivo
+\series default
+: 
+\begin_inset Formula $\vec{v}\neq\vec{0}\implies\vec{v}\cdot\vec{v}>0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+espacio vectorial euclídeo
+\series default
+ es un espacio vectorial real en el que hay definido un producto escalar.
+ Todo subespacio vectorial suyo es también euclídeo.
+ Ejemplos:
+\end_layout
+
+\begin_layout Itemize
+El 
+\series bold
+producto escalar usual
+\series default
+ en 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ viene dado por 
+\begin_inset Formula $\vec{v}\cdot\vec{w}=\sum_{i=1}^{n}x_{i}y_{i}$
+\end_inset
+
+, es decir, 
+\begin_inset Formula 
+\[
+\vec{v}\cdot\vec{w}=\left(\begin{array}{ccc}
+- & \vec{v} & -\end{array}\right)\left(\begin{array}{c}
+|\\
+\vec{w}\\
+|
+\end{array}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+El 
+\series bold
+producto escalar integral
+\series default
+ en el espacio 
+\begin_inset Formula ${\cal C}[a,b]$
+\end_inset
+
+ de las funciones reales continuas en el intervalo 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+, o en sus subespacios 
+\begin_inset Formula ${\cal P}[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal P}_{n}[a,b]$
+\end_inset
+
+ de funciones polinómicas arbitrarias y de grado máximo 
+\begin_inset Formula $n$
+\end_inset
+
+, respectivamente, viene dado por
+\begin_inset Formula 
+\[
+f\cdot g=\int_{a}^{b}f(x)g(x)dx
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Norma y coseno
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+norma
+\series default
+, 
+\series bold
+módulo
+\series default
+ o 
+\series bold
+longitud
+\series default
+ de un vector 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ es 
+\begin_inset Formula $\Vert\vec{v}\Vert=\sqrt{\vec{v}\cdot\vec{v}}$
+\end_inset
+
+, y 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ es 
+\series bold
+unitario
+\series default
+ si 
+\begin_inset Formula $\Vert\vec{v}\Vert=1$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert\vec{v}\Vert=0\iff\vec{v}=\vec{0}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert r\vec{v}\Vert=|r|\Vert\vec{v}\Vert$
+\end_inset
+
+, y en particular 
+\begin_inset Formula $\frac{\vec{v}}{\Vert\vec{v}\Vert}$
+\end_inset
+
+ es unitario.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Teorema del coseno
+\series default
+: 
+\begin_inset Formula $\Vert\vec{v}\pm\vec{w}\Vert^{2}=\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}\pm2\vec{v}\cdot\vec{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Desigualdad de Cauchy-Schwartz
+\series default
+: 
+\begin_inset Formula $|\vec{v}\cdot\vec{w}|\leq\Vert\vec{v}\Vert\Vert\vec{w}\Vert$
+\end_inset
+
+, y la igualdad se cumple si y sólo si no son proporcionales.
+\begin_inset Newline newline
+\end_inset
+
+Si 
+\begin_inset Formula $\vec{v}=0$
+\end_inset
+
+ es trivial.
+ Si no, para cada 
+\begin_inset Formula $x\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $0\leq\Vert x\vec{v}-\vec{w}\Vert^{2}=\Vert\vec{v}\Vert^{2}x^{2}-(2\vec{v}\cdot\vec{w})x+\Vert\vec{w}\Vert^{2}$
+\end_inset
+
+.
+ Luego tenemos un polinomio de 
+\begin_inset Formula $2^{o}$
+\end_inset
+
+ grado con a lo más una raíz real (pues 
+\begin_inset Formula $\Vert x\vec{v}-\vec{w}\Vert^{2}=0\iff x\vec{v}-\vec{w}=0$
+\end_inset
+
+), de modo que el discriminante 
+\begin_inset Formula $4(\vec{v}\cdot\vec{w})^{2}-4\Vert\vec{v}\Vert^{2}\Vert\vec{w}\Vert^{2}$
+\end_inset
+
+ no puede ser estrictamente positivo, es decir, debe ser 
+\begin_inset Formula $|\vec{v}\cdot\vec{w}|\leq\Vert\vec{v}\Vert\Vert\vec{w}\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+
+\series bold
+Desigualdades de Minkowski y triangular
+\series default
+: 
+\begin_inset Formula $\Vert\vec{v}\Vert-\Vert\vec{w}\Vert\leq\Vert\vec{v}\pm\vec{w}\Vert\leq\Vert\vec{v}\Vert+\Vert\vec{w}\Vert$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Tomando cuadrados, 
+\begin_inset Formula $\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}-2\Vert\vec{v}\Vert\Vert\vec{w}\Vert\leq\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}\pm2\vec{v}\cdot\vec{w}\leq\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}+2\Vert\vec{v}\Vert\Vert\vec{w}\Vert$
+\end_inset
+
+, y cancelando 
+\begin_inset Formula $\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}$
+\end_inset
+
+ y aplicando Cauchy-Schwartz tenemos el resultado.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+coseno
+\series default
+ del ángulo formado por dos vectores 
+\begin_inset Formula $\vec{v},\vec{w}\neq\vec{0}$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+\cos(\vec{v},\vec{w}):=\frac{\vec{v}\cdot\vec{w}}{\Vert\vec{v}\Vert\Vert\vec{w}\Vert}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dos vectores 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ son 
+\series bold
+ortogonales
+\series default
+ o 
+\series bold
+perpendiculares
+\series default
+ (
+\begin_inset Formula $\vec{v}\bot\vec{w}$
+\end_inset
+
+) si 
+\begin_inset Formula $\vec{v}\cdot\vec{w}=0$
+\end_inset
+
+.
+ Así, 
+\begin_inset Formula $\vec{0}$
+\end_inset
+
+ es ortogonal a todos y 
+\begin_inset Formula $\vec{v},\vec{w}\neq\vec{0}$
+\end_inset
+
+ son ortogonales si y sólo si 
+\begin_inset Formula $\cos(\vec{v},\vec{w})=0$
+\end_inset
+
+.
+ Del teorema del coseno se deduce el 
+\series bold
+teorema de Pitágoras
+\series default
+:
+\begin_inset Formula 
+\[
+\vec{v}\bot\vec{w}\iff\Vert\vec{v}+\vec{w}\Vert^{2}=\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Conjuntos ortogonales
+\end_layout
+
+\begin_layout Standard
+Se dice que 
+\begin_inset Formula $\vec{x}\in V$
+\end_inset
+
+ es ortogonal al subespacio 
+\begin_inset Formula $U$
+\end_inset
+
+ si lo es a todos los vectores de 
+\begin_inset Formula $U$
+\end_inset
+
+, o por linealidad a los de un conjunto generador de 
+\begin_inset Formula $U$
+\end_inset
+
+ cualquiera.
+ Llamamos 
+\series bold
+subespacio ortogonal
+\series default
+ de 
+\begin_inset Formula $U$
+\end_inset
+
+ en 
+\begin_inset Formula $V$
+\end_inset
+
+, escrito 
+\begin_inset Formula $U^{\bot}$
+\end_inset
+
+, al conjunto de todos los vectores de 
+\begin_inset Formula $V$
+\end_inset
+
+ ortogonales a 
+\begin_inset Formula $U$
+\end_inset
+
+, que por la linealidad del producto escalar es un subespacio (incluso aunque
+ 
+\begin_inset Formula $U$
+\end_inset
+
+ no lo sea).
+ Sólo el vector nulo es ortogonal a sí mismo, luego 
+\begin_inset Formula $U\cap U^{\bot}=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dos subespacios 
+\begin_inset Formula $U$
+\end_inset
+
+ y 
+\begin_inset Formula $W$
+\end_inset
+
+ son 
+\series bold
+ortogonales
+\series default
+ si 
+\begin_inset Formula $\forall\vec{u}\in U,\vec{w}\in W;\vec{u}\bot\vec{w}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\dim(U)+\dim(W)>\dim(V)$
+\end_inset
+
+, diremos que 
+\begin_inset Formula $U$
+\end_inset
+
+ y 
+\begin_inset Formula $W$
+\end_inset
+
+ son ortogonales cuando lo sean 
+\begin_inset Formula $U^{\bot}$
+\end_inset
+
+ y 
+\begin_inset Formula $W^{\bot}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $U+W=V$
+\end_inset
+
+, diremos que 
+\begin_inset Formula $W$
+\end_inset
+
+ es un 
+\series bold
+complemento ortogonal
+\series default
+ de 
+\begin_inset Formula $U$
+\end_inset
+
+ (o al revés).
+\end_layout
+
+\begin_layout Standard
+Un conjunto de vectores en un espacio euclídeo 
+\begin_inset Formula $V$
+\end_inset
+
+ es 
+\series bold
+ortogonal
+\series default
+ si sus vectores son no nulos y ortogonales dos a dos, y es 
+\series bold
+ortonormal
+\series default
+ si además son unitarios.
+ Si en un conjunto ortogonal dividimos cada vector por su norma, nos queda
+ un conjunto ortonormal que genera el mismo subespacio.
+\end_layout
+
+\begin_layout Standard
+Todo conjunto ortogonal 
+\begin_inset Formula $\{\vec{u}_{1},\dots,\vec{u}_{m}\}$
+\end_inset
+
+ es linealmente independiente.
+ 
+\series bold
+Demostración:
+\series default
+ Si no lo fuera, habría un vector combinación lineal del resto, por ejemplo,
+ 
+\begin_inset Formula $\vec{u}_{1}=a_{2}\vec{u}_{2}+\dots+a_{m}\vec{u}_{m}$
+\end_inset
+
+, y se tendría que
+\begin_inset Formula 
+\[
+0\neq\vec{u}_{1}\cdot\vec{u}_{1}=\vec{u}_{1}\cdot(a_{2}\vec{u}_{2}+\dots+a_{m}\vec{u}_{m})=a_{2}(\vec{u}_{1}\cdot\vec{u}_{2})+\dots+a_{m}(\vec{u}_{1}\cdot\vec{u}_{m})=0\#
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Por esto también hablamos de 
+\series bold
+bases ortogonales
+\series default
+ u 
+\series bold
+ortonormales
+\series default
+.
+ Por ejemplo, en 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+, la base canónica es una base ortonormal.
+\end_layout
+
+\begin_layout Standard
+Una matriz 
+\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{R})$
+\end_inset
+
+ es 
+\series bold
+ortogonal
+\series default
+ si 
+\begin_inset Formula $A^{t}=A^{-1}$
+\end_inset
+
+, si y sólo si sus columnas (o filas) forman una base ortonormal de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración: 
+\series default
+Si 
+\begin_inset Formula $\vec{u}_{1},\dots,\vec{u}_{n}$
+\end_inset
+
+ son vectores no nulos de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $A$
+\end_inset
+
+ es la matriz que tiene por columnas estos vectores, 
+\begin_inset Formula $A^{t}A$
+\end_inset
+
+ es una matriz cuadrada 
+\begin_inset Formula $n\times n$
+\end_inset
+
+ con 
+\begin_inset Formula $(A^{t}A)_{ij}=\vec{u}_{i}\cdot\vec{u}_{j}$
+\end_inset
+
+, luego los vectores son ortogonales si y sólo si 
+\begin_inset Formula $A^{t}A$
+\end_inset
+
+ es diagonal (sin ceros en la diagonal), y son ortonormales si y sólo si
+ 
+\begin_inset Formula $A^{t}A=I_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Método de Gram-Schmidt
+\end_layout
+
+\begin_layout Standard
+Dado un conjunto ortogonal 
+\begin_inset Formula $\{\vec{u}_{1},\dots,\vec{u}_{k}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{x}\notin U=<\vec{u}_{1},\dots,\vec{u}_{k}>$
+\end_inset
+
+, el vector 
+\begin_inset Formula $\vec{u}_{k+1}:=\vec{x}-\frac{\vec{x}\cdot\vec{u}_{1}}{\Vert\vec{u}_{1}\Vert^{2}}\vec{u}_{1}-\dots-\frac{\vec{x}\cdot\vec{u}_{k}}{\Vert\vec{u}_{k}\Vert^{2}}\vec{u}_{k}$
+\end_inset
+
+ es ortogonal a los del conjunto y 
+\begin_inset Formula $<\vec{u}_{1},\dots,\vec{u}_{k},\vec{u}_{k+1}>=<\vec{u}_{1},\dots,\vec{u}_{k},\vec{x}>$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ El que ambos generen el mismo subespacio es consecuencia de que 
+\begin_inset Formula $\vec{u}_{k+1}-\vec{x}\in<\vec{u}_{1},\dots,\vec{u}_{k}>$
+\end_inset
+
+.
+ Además, dado 
+\begin_inset Formula $j\in\{1,\dots,k\}$
+\end_inset
+
+, 
+\begin_inset Formula $\vec{u}_{k+1}\cdot\vec{u}_{j}=\vec{x}\cdot\vec{u}_{j}-\sum_{i=1}^{k}\frac{\vec{x}\cdot\vec{u}_{i}}{\Vert\vec{u}_{i}\Vert^{2}}\vec{u}_{i}\cdot\vec{u}_{j}=\vec{x}\cdot\vec{u}_{j}-\frac{\vec{x}\cdot\vec{u}_{j}}{\Vert\vec{u}_{j}\Vert^{2}}\vec{u}_{j}\cdot\vec{u}_{j}=0$
+\end_inset
+
+, luego 
+\begin_inset Formula $\vec{u}_{k+1}\bot U$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+De aquí que todo subespacio 
+\begin_inset Formula $U=\{\vec{x}_{1},\dots,\vec{x}_{m}\}$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ admite una base ortogonal 
+\begin_inset Formula $\{\vec{u}_{1},\dots,\vec{u}_{m}\}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $<\vec{x}_{1}>=<\vec{u}_{1}>,\dots,<\vec{x}_{1},\dots,\vec{x}_{m}>=<\vec{u}_{1},\dots,\vec{u}_{m}>$
+\end_inset
+
+.
+ Podemos obtener esta base por el 
+\series bold
+algoritmo de ortogonalización de Gram-Schmidt
+\series default
+: Tomamos 
+\begin_inset Formula $\vec{u}_{1}=\vec{x}_{1}$
+\end_inset
+
+ y, para cada 
+\begin_inset Formula $j\in\{1,\dots,m\}$
+\end_inset
+
+, 
+\begin_inset Formula $\vec{u}_{j}=\vec{x}_{j}-\sum_{i=1}^{j-1}\frac{\vec{x}_{j}\cdot\vec{u}_{i}}{\Vert\vec{u}_{i}\Vert^{2}}\vec{u}_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Por tanto, todo subespacio 
+\begin_inset Formula $U$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ tiene una base ortonormal, que podemos ampliar a una base ortonormal de
+ 
+\begin_inset Formula $V$
+\end_inset
+
+, y los vectores añadidos son una base ortonormal de 
+\begin_inset Formula $U^{\bot}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $U\oplus U^{\bot}=V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+De aquí que, si 
+\begin_inset Formula $U$
+\end_inset
+
+ y 
+\begin_inset Formula $W$
+\end_inset
+
+ son subespacios de un espacio vectorial euclídeo 
+\begin_inset Formula $V$
+\end_inset
+
+ de dimensión finita, entonces 
+\begin_inset Formula $(U^{\bot})^{\bot}=U$
+\end_inset
+
+, 
+\begin_inset Formula $U\subseteq W\iff W^{\bot}\subseteq U^{\bot}$
+\end_inset
+
+, 
+\begin_inset Formula $U^{\bot}\cap W^{\bot}=(U+W)^{\bot}$
+\end_inset
+
+ y 
+\begin_inset Formula $U^{\bot}+W^{\bot}=(U\cap W)^{\bot}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+En 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ con el producto escalar usual, si 
+\begin_inset Formula $U$
+\end_inset
+
+ está generado por las filas de la matriz 
+\begin_inset Formula $A$
+\end_inset
+
+ entonces 
+\begin_inset Formula $U^{\bot}=\text{Nuc}(A)$
+\end_inset
+
+, y viceversa.
+\end_layout
+
+\begin_layout Subsection
+Coeficientes de Fourier y proyección ortogonal
+\end_layout
+
+\begin_layout Standard
+Dados 
+\begin_inset Formula ${\cal B}=\{\vec{u}_{1},\dots,\vec{u}_{m}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{x}\in V$
+\end_inset
+
+, los 
+\series bold
+coeficientes de Fourier
+\series default
+ de 
+\begin_inset Formula $\vec{x}$
+\end_inset
+
+ en 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ son los escalares 
+\begin_inset Formula $r_{i}=\frac{\vec{x}\cdot\vec{u}_{i}}{\Vert\vec{u}_{i}\Vert^{2}}$
+\end_inset
+
+ para 
+\begin_inset Formula $i\in\{1,\dots,m\}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\vec{x}\in<{\cal B}>$
+\end_inset
+
+, estas son sus coordenadas respecto a la base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+, pues 
+\begin_inset Formula $\vec{x}\cdot\vec{u}_{i}=\left(\sum_{j=1}^{m}r_{j}\vec{u}_{j}\right)\cdot\vec{u}_{i}=\sum_{j=1}^{m}r_{j}(\vec{u}_{j}\cdot\vec{u}_{i})=r_{i}\Vert\vec{u}_{i}\Vert^{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+proyección ortogonal
+\series default
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ sobre 
+\begin_inset Formula $U$
+\end_inset
+
+ a la aplicación lineal 
+\begin_inset Formula $\pi_{U}:V=U\oplus U^{\bot}\rightarrow U$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $\vec{v}=\vec{v}_{1}+\vec{v}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}_{1}\in U$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v}_{2}\in U^{\bot}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\pi_{U}(\vec{v})=\vec{v}_{1}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $r_{1},\dots,r_{m}$
+\end_inset
+
+ son los coeficientes de Fourier de 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ sobre la base ortogonal 
+\begin_inset Formula $\{\vec{u}_{1},\dots,\vec{u}_{m}\}$
+\end_inset
+
+ de 
+\begin_inset Formula $U$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\pi_{U}(\vec{v})=\sum_{i=1}^{m}r_{i}\vec{u}_{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\vec{u}:=\pi_{U}(\vec{v})$
+\end_inset
+
+ es la 
+\series bold
+mejor aproximación
+\series default
+ de 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ en 
+\begin_inset Formula $U$
+\end_inset
+
+, es decir, 
+\begin_inset Formula $\min\{\Vert\vec{v}-\vec{z}\Vert\}_{\vec{z}\in U}=\Vert\vec{v}-\vec{u}\Vert$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $\vec{w}=\vec{v}-\vec{u}$
+\end_inset
+
+, si 
+\begin_inset Formula $\vec{z}\in U$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\vec{u}-\vec{z}\bot\vec{w}$
+\end_inset
+
+, y por el teorema de Pitágoras, 
+\begin_inset Formula $\Vert\vec{v}-\vec{z}\Vert=\Vert\vec{w}+\vec{u}-\vec{z}\Vert=\sqrt{\Vert\vec{w}\Vert^{2}+\Vert\vec{u}-\vec{z}\Vert^{2}}$
+\end_inset
+
+, con lo que el valor mínimo de 
+\begin_inset Formula $\Vert\vec{v}-\vec{z}\Vert$
+\end_inset
+
+ es 
+\begin_inset Formula $\Vert\vec{w}\Vert$
+\end_inset
+
+ y se alcanza cuando 
+\begin_inset Formula $\vec{z}=\vec{u}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+simetría ortogonal
+\series default
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ sobre 
+\begin_inset Formula $U$
+\end_inset
+
+ es la aplicación lineal 
+\begin_inset Formula $\sigma_{U}:V\rightarrow V$
+\end_inset
+
+ con 
+\begin_inset Formula $\sigma_{U}(\vec{v})=\vec{v}_{1}-\vec{v}_{2}=2\pi_{U}(\vec{v})-\vec{v}$
+\end_inset
+
+ para todo 
+\begin_inset Formula $\vec{v}\in V$
+\end_inset
+
+, siendo 
+\begin_inset Formula $\vec{v}=\vec{v}_{1}+\vec{v}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}_{1}\in U$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v}_{2}\in U^{\bot}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Productos vectorial y mixto
+\end_layout
+
+\begin_layout Standard
+En 
+\begin_inset Formula $\mathbb{R}^{3}$
+\end_inset
+
+, el 
+\series bold
+producto vectorial
+\series default
+ de 
+\begin_inset Formula $\vec{v}=(v_{1},v_{2},v_{3})$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}=(w_{1},w_{2},w_{3})$
+\end_inset
+
+ es el vector
+\begin_inset Formula 
+\[
+\vec{v}\land\vec{w}:=\left|\begin{array}{ccc}
+\vec{e}_{1} & v_{1} & w_{1}\\
+\vec{e}_{2} & v_{2} & w_{2}\\
+\vec{e}_{3} & v_{3} & w_{3}
+\end{array}\right|
+\]
+
+\end_inset
+
+y el 
+\series bold
+producto mixto
+\series default
+ de 
+\begin_inset Formula $\vec{u}$
+\end_inset
+
+, 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ es el escalar 
+\begin_inset Formula $\vec{u}\cdot(\vec{v}\land\vec{w})$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\vec{u}\cdot(\vec{v}\land\vec{w})=\det(\vec{u},\vec{v},\vec{w})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\vec{v}\land\vec{w}=-(\vec{w}\land\vec{v})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\vec{v}\land(\vec{w}_{1}+\mu\vec{w}_{2})=\vec{v}\land\vec{w}_{1}+\mu\vec{v}\land\vec{w}_{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ son linealmente independientes, 
+\begin_inset Formula $\vec{v}\land\vec{w}$
+\end_inset
+
+ es perpendicular a ambos, por lo que genera la recta ortogonal al plano
+ que determinan: 
+\begin_inset Formula $<\vec{v}\land\vec{w}>=<\vec{v},\vec{w}>^{\bot}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\vec{v}\cdot(\vec{v}\land\vec{w})=\vec{w}\cdot(\vec{v}\land\vec{w})=\vec{0}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\Vert\vec{v}\land\vec{w}\Vert^{2}+(\vec{v}\cdot\vec{w})^{2}=\Vert\vec{v}\Vert^{2}\Vert\vec{w}\Vert^{2}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\left(\frac{\Vert\vec{v}\land\vec{w}\Vert}{\Vert\vec{v}\Vert\Vert\vec{w}\Vert}\right)^{2}+\left(\frac{\vec{v}\cdot\vec{w}}{\Vert\vec{v}\Vert\Vert\vec{w}\Vert}\right)^{2}=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Por la última propiedad, el 
+\series bold
+seno
+\series default
+ del ángulo que forman 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ cumple que
+\begin_inset Formula 
+\[
+|\sin(\vec{v},\vec{w})|=\frac{\Vert\vec{v}\land\vec{w}\Vert}{\Vert\vec{v}\Vert\Vert\vec{w}\Vert}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Por tanto, el 
+\series bold
+área del paralelogramo
+\series default
+ dado por 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ es 
+\begin_inset Formula $\Vert\vec{x}\land\vec{z}\Vert=\Vert\vec{x}\Vert(\Vert\vec{z}\Vert|\sin(\vec{x},\vec{z})|)$
+\end_inset
+
+, y el 
+\series bold
+volumen del paralelepípedo
+\series default
+ determinado por 
+\begin_inset Formula $\vec{u}$
+\end_inset
+
+, 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}$
+\end_inset
+
+ es 
+\begin_inset Formula $|\vec{u}\cdot(\vec{v}\land\vec{w})|=\Vert\vec{v}\land\vec{w}\Vert(\Vert\vec{u}\Vert|\cos(\vec{v}\land\vec{w},\vec{u})|)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Espacios afines euclídeos
+\end_layout
+
+\begin_layout Standard
+Un 
+\series bold
+espacio afín euclídeo
+\series default
+ es un espacio afín 
+\begin_inset Formula $E$
+\end_inset
+
+ cuyo espacio vectorial asociado 
+\begin_inset Formula $V$
+\end_inset
+
+ es euclídeo.
+ Si 
+\begin_inset Formula $V$
+\end_inset
+
+ tiene dimensión finita, llamamos 
+\series bold
+sistema de referencia ortonormal
+\series default
+ o 
+\series bold
+referencial ortonormal
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+ a un referencial cartesiano 
+\begin_inset Formula $\Re=(O,{\cal B})$
+\end_inset
+
+ en el que 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ es base ortonormal de 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Denotamos con 
+\begin_inset Formula $E$
+\end_inset
+
+ un espacio afín euclídeo de dimensión finita y 
+\begin_inset Formula $E_{n}$
+\end_inset
+
+ a 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ con su estructura afín y euclídea estándar.
+\end_layout
+
+\begin_layout Standard
+Definimos la 
+\series bold
+distancia
+\series default
+ entre dos puntos 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ como 
+\begin_inset Formula $d(P,Q):=\Vert\overrightarrow{PQ}\Vert$
+\end_inset
+
+, y por las propiedades de la norma, 
+\begin_inset Formula $d(P,Q)\geq0$
+\end_inset
+
+ con 
+\begin_inset Formula $d(P,Q)=0\iff P=Q$
+\end_inset
+
+, 
+\begin_inset Formula $d(P,Q)=d(Q,P)$
+\end_inset
+
+ y 
+\begin_inset Formula $d(P,R)\leq d(P,Q)+d(Q,R)$
+\end_inset
+
+, por lo que se trata de una métrica.
+ En particular, si 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ tienen coordenadas 
+\begin_inset Formula $(p_{1},\dots,p_{n})$
+\end_inset
+
+ y 
+\begin_inset Formula $(q_{1},\dots,q_{n})$
+\end_inset
+
+ en un referencial ortonormal, entonces
+\begin_inset Formula 
+\[
+d(P,Q)=\sqrt{\sum_{i=1}^{n}(p_{i}-q_{i})^{2}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+La distancia entre dos variedades 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}'$
+\end_inset
+
+ se define como 
+\begin_inset Formula $d({\cal L},{\cal L}'):=\inf\{d(P,P')\}_{P\in{\cal L},P'\in{\cal L}'}$
+\end_inset
+
+, y la distancia de un punto 
+\begin_inset Formula $Q$
+\end_inset
+
+ a una variedad 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ como 
+\begin_inset Formula $d(Q,{\cal L})=\inf\{d(P,Q)\}_{P\in{\cal L}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dos variedades 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}'$
+\end_inset
+
+ son 
+\series bold
+ortogonales
+\series default
+ o 
+\series bold
+perpendiculares
+\series default
+ (
+\begin_inset Formula ${\cal L}\bot{\cal L}'$
+\end_inset
+
+) si lo son sus direcciones, y llamamos 
+\series bold
+variedad perpendicular
+\series default
+ a 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ que pasa por 
+\begin_inset Formula $Q$
+\end_inset
+
+ a la variedad 
+\begin_inset Formula $Q+W^{\bot}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así, si 
+\begin_inset Formula $\ell_{1}=P_{1}+<\vec{v}_{1}>$
+\end_inset
+
+ y 
+\begin_inset Formula $\ell_{2}=P_{2}+<\vec{v}_{2}>$
+\end_inset
+
+ son rectas en 
+\begin_inset Formula $E_{3}$
+\end_inset
+
+ que se cruzan, sea 
+\begin_inset Formula $\vec{v}_{3}=\vec{v}_{1}\land\vec{v}_{2}$
+\end_inset
+
+ la dirección perpendicular a ambas, como 
+\begin_inset Formula $\vec{v}_{3}\notin<\vec{v}_{1},\vec{v}_{2}>$
+\end_inset
+
+, existe una única recta con esta dirección que corte a 
+\begin_inset Formula $\ell_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\ell_{2}$
+\end_inset
+
+, que llamamos 
+\series bold
+perpendicular común
+\series default
+ de ambas.
+ Para calcularla, hallamos el punto 
+\begin_inset Formula $Q\in\ell_{1}\cap(P_{2}+<\vec{v}_{2},\vec{v}_{3}>)$
+\end_inset
+
+ y tomamos la recta 
+\begin_inset Formula $Q+<\vec{v}_{3}>$
+\end_inset
+
+, o buscamos 
+\begin_inset Formula $\lambda_{1},\lambda_{2},\lambda_{3}\in\mathbb{R}$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\overrightarrow{(P_{1}+\lambda_{1}\vec{v}_{1})(P_{2}+\lambda_{2}\vec{v}_{2})}=\lambda_{3}\vec{v}_{3}$
+\end_inset
+
+, es decir, tales que 
+\begin_inset Formula $\overrightarrow{P_{1}P_{2}}=\lambda_{1}\vec{v}_{1}-\lambda_{2}\vec{v}_{2}+\lambda_{3}\vec{v}_{3}$
+\end_inset
+
+, y tomamos la recta 
+\begin_inset Formula $(P_{1}+\lambda_{1}\vec{v}_{1})+<\vec{v}_{3}>$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dados un punto 
+\begin_inset Formula $Q$
+\end_inset
+
+ y una variedad 
+\begin_inset Formula ${\cal L}=P+W$
+\end_inset
+
+ en 
+\begin_inset Formula $E$
+\end_inset
+
+, definimos la 
+\series bold
+proyección ortogonal
+\series default
+ de 
+\begin_inset Formula $Q$
+\end_inset
+
+ sobre 
+\begin_inset Formula ${\cal L}$
+\end_inset
+
+ como el único punto 
+\begin_inset Formula $Q'\in{\cal L}\cap(Q+W^{\bot})$
+\end_inset
+
+, y el 
+\series bold
+simétrico ortogonal
+\series default
+ como el punto 
+\begin_inset Formula $Q''=Q+2\overrightarrow{QQ'}$
+\end_inset
+
+.
+ Con esto, 
+\begin_inset Formula $d(Q,{\cal L})=d(Q,Q')$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $Q'\in{\cal L}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{QQ'}\in W^{\bot}$
+\end_inset
+
+, luego para un 
+\begin_inset Formula $X\in{\cal L}$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $Q',X\in{\cal L}\implies\overrightarrow{Q'X}\in W\implies\overrightarrow{QQ'}\bot\overrightarrow{Q'X}\implies d(Q,X)=\sqrt{d(Q,Q')^{2}+d(Q',X)^{2}}$
+\end_inset
+
+, con lo que el mínimo se alcanza en 
+\begin_inset Formula $X=Q'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Itemize
+La distancia de un punto 
+\begin_inset Formula $Q=(q_{1},\dots,q_{n})$
+\end_inset
+
+ a un hiperplano 
+\begin_inset Formula ${\cal H}$
+\end_inset
+
+ de ecuación 
+\begin_inset Formula $a_{1}x_{1}+\dots+a_{n}x_{n}+b=0$
+\end_inset
+
+ es 
+\begin_inset Formula $d(Q,{\cal H})=\frac{|a_{1}q_{1}+\dots+a_{n}q_{n}+b|}{\Vert(a_{1},\dots,a_{n})\Vert}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+La recta ortogonal a 
+\begin_inset Formula ${\cal H}$
+\end_inset
+
+ por 
+\begin_inset Formula $Q$
+\end_inset
+
+ es 
+\begin_inset Formula $Q+<\vec{a}>$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{a}=(a_{1},\dots,a_{n})$
+\end_inset
+
+, y sus puntos tienen la forma 
+\begin_inset Formula $(q_{1}+\lambda a_{1},\dots,q_{n}+\lambda a_{n})$
+\end_inset
+
+.
+ Para cierto 
+\begin_inset Formula $\lambda_{0}$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $Q':=Q+\lambda_{0}\vec{a}\in{\cal H}$
+\end_inset
+
+.
+ Sustituyendo, 
+\begin_inset Formula $0=a_{1}(q_{1}+\lambda_{0}a_{1})+\dots+a_{n}(q_{n}+\lambda_{0}a_{n})+b=a_{1}q_{1}+\dots+a_{n}q_{n}+b+\lambda_{0}\Vert\vec{a}\Vert^{2}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\lambda_{0}=-\frac{a_{1}q_{1}+\dots+a_{n}q_{n}+b}{\Vert\vec{a}\Vert^{2}}$
+\end_inset
+
+, y la fórmula se obtiene de que 
+\begin_inset Formula $d(Q,Q')=|\lambda_{0}|\Vert\vec{a}\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+La distancia de un punto 
+\begin_inset Formula $Q$
+\end_inset
+
+ a una recta 
+\begin_inset Formula $\ell=P+<\vec{v}=(v_{1},v_{2})>$
+\end_inset
+
+ en 
+\begin_inset Formula $E_{2}$
+\end_inset
+
+ es 
+\begin_inset Formula $d(Q,\ell)=\frac{|\det(\overrightarrow{PQ},\vec{v})|}{\Vert\vec{v}\Vert}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+La ecuación implícita de la recta es 
+\begin_inset Formula $\det(\overrightarrow{PX},\vec{v})=0$
+\end_inset
+
+, cuyos coeficientes, 
+\begin_inset Formula $(-v_{2},v_{1})$
+\end_inset
+
+, tienen la misma norma que 
+\begin_inset Formula $\Vert\vec{v}\Vert$
+\end_inset
+
+, con lo que la fórmula se deduce del ejemplo anterior.
+\end_layout
+
+\begin_layout Itemize
+La distancia de un punto 
+\begin_inset Formula $Q$
+\end_inset
+
+ a un plano 
+\begin_inset Formula $\pi=P+<\vec{v},\vec{w}>$
+\end_inset
+
+ es 
+\begin_inset Formula $d(Q,\pi)=\frac{|\det(\overrightarrow{PQ},\vec{v},\vec{w})|}{\Vert\vec{v}\land\vec{w}\Vert}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+La ecuación implícita del plano es 
+\begin_inset Formula $\det(\overrightarrow{PX},\vec{v},\vec{w})=0$
+\end_inset
+
+ , cuyos coeficientes son los del vector ortogonal al plano, 
+\begin_inset Formula $\vec{v}\land\vec{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+La distancia de un punto 
+\begin_inset Formula $Q$
+\end_inset
+
+ a una recta 
+\begin_inset Formula $\ell=P+<\vec{v}>$
+\end_inset
+
+ es 
+\begin_inset Formula $d(Q,\ell)=\frac{\Vert\vec{v}\land\overrightarrow{PQ}\Vert}{\Vert\vec{v}\Vert}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Si 
+\begin_inset Formula $Q'$
+\end_inset
+
+ es la proyección ortogonal de 
+\begin_inset Formula $Q$
+\end_inset
+
+ sobre 
+\begin_inset Formula $\ell$
+\end_inset
+
+, se tiene 
+\begin_inset Formula $d(Q,\ell)=\Vert\overrightarrow{Q'Q}\Vert$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{PQ}=\overrightarrow{PQ'}+\overrightarrow{Q'Q}$
+\end_inset
+
+ con 
+\begin_inset Formula $\overrightarrow{PQ'}$
+\end_inset
+
+ proporcional a 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{Q'Q}\bot\vec{v}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\vec{v}\land\overrightarrow{PQ}=\vec{v}\land\overrightarrow{Q'Q}$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\Vert\vec{v}\land\overrightarrow{PQ}\Vert=\Vert\vec{v}\Vert\Vert\overrightarrow{Q'Q}\Vert$
+\end_inset
+
+, de donde se deduce la fórmula.
+\end_layout
+
+\begin_layout Standard
+Dadas 
+\begin_inset Formula ${\cal L}_{1}=P_{1}+W_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal L}_{2}=P_{2}+W_{2}$
+\end_inset
+
+, 
+\begin_inset Formula $d({\cal L}_{1},{\cal L}_{2})=d(P_{1},P_{2}+(W_{1}+W_{2}))$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Veamos que los conjuntos 
+\begin_inset Formula $A=\{d(X_{1},X_{2})\}_{X_{1}\in{\cal L}_{1},X_{2}\in{\cal L}_{2}}$
+\end_inset
+
+ y 
+\begin_inset Formula $B=\{d(P_{1},X)\}_{X\in P_{2}+(W_{1}+W_{2})}$
+\end_inset
+
+ son iguales.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\subseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $X_{1}=P_{1}+\vec{w}_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $X_{2}=P_{2}+\vec{w}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{w}_{1}\in W_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}_{2}\in W_{2}$
+\end_inset
+
+, entonces 
+\begin_inset Formula 
+\[
+d(X_{1},X_{2})=\Vert\overrightarrow{X_{1}X_{2}}\Vert=\Vert\overrightarrow{(P_{1}+\vec{w}_{1})(P_{2}+\vec{w}_{2})}\Vert=\Vert\overrightarrow{P_{1}(P_{2}+\vec{w}_{2}-\vec{w}_{1})}\Vert\in B
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\supseteq]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dado 
+\begin_inset Formula $X=P_{2}+\vec{w}_{1}+\vec{w}_{2}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{w}_{1}\in W_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}_{2}\in W_{2}$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+d(P_{1},X)=\Vert\overrightarrow{P_{1}(P_{2}+\vec{w}_{1}+\vec{w}_{2})}\Vert=\Vert\overrightarrow{(P_{1}-\vec{w}_{1})(P_{2}+\vec{w}_{2})}\Vert\in A
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/gae/n4.lyx b/gae/n4.lyx
new file mode 100644
index 0000000..8e02e9a
--- /dev/null
+++ b/gae/n4.lyx
@@ -0,0 +1,1509 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
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+\font_roman "default" "default"
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+\use_microtype false
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+\graphics default
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+\output_sync 0
+\bibtex_command default
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+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
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+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Una 
+\series bold
+transformación ortogonal
+\series default
+ de un espacio vectorial euclídeo 
+\begin_inset Formula $V$
+\end_inset
+
+ es una aplicación 
+\begin_inset Formula $f:V\rightarrow V$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\vec{v}\cdot\vec{w}=f(\vec{v})\cdot f(\vec{w})$
+\end_inset
+
+ para cualesquiera 
+\begin_inset Formula $\vec{v},\vec{w}\in V$
+\end_inset
+
+, y el conjunto de estas transformaciones se conoce como 
+\series bold
+grupo ortogonal
+\series default
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ (
+\begin_inset Formula ${\cal O}(V)$
+\end_inset
+
+).
+ Si la aplicación es entre espacios distintos hablamos de una 
+\series bold
+aplicación ortogonal
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Una aplicación 
+\begin_inset Formula $f:V\rightarrow V$
+\end_inset
+
+ es una transformación ortogonal si y sólo si es lineal y conserva normas.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si se conservan productos escalares se conservan normas.
+ Sean 
+\begin_inset Formula $r\in\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{v},\vec{w}\in V$
+\end_inset
+
+.
+ Para ver que 
+\begin_inset Formula $f(r\vec{v})=rf(\vec{v})$
+\end_inset
+
+, vemos que
+\begin_inset Formula 
+\begin{multline*}
+\Vert f(r\vec{v})-rf(\vec{v})\Vert^{2}=\Vert f(r\vec{v})\Vert^{2}+\Vert rf(\vec{v})\Vert^{2}-2f(r\vec{v})\cdot(rf(\vec{v}))=\\
+=\Vert r\vec{v}\Vert^{2}+r^{2}\Vert f(\vec{v})\Vert^{2}-2r(f(r\vec{v})\cdot f(\vec{v}))=r^{2}\Vert\vec{v}\Vert^{2}+r^{2}\Vert\vec{v}\Vert^{2}-2r(r\vec{v}\cdot\vec{v})=0
+\end{multline*}
+
+\end_inset
+
+Para ver que 
+\begin_inset Formula $f(\vec{v}+\vec{w})=f(\vec{v})+f(\vec{w})$
+\end_inset
+
+,
+\begin_inset Formula 
+\begin{multline*}
+\Vert f(\vec{v}+\vec{w})-f(\vec{v})-f(\vec{w})\Vert^{2}=\\
+=\Vert f(\vec{v}+\vec{w})\Vert^{2}+\Vert f(\vec{v})\Vert^{2}+\Vert f(\vec{w})\Vert^{2}+2(f(\vec{v})\cdot f(\vec{w})-f(\vec{v}+\vec{w})\cdot f(\vec{v})-f(\vec{v}+\vec{w})\cdot f(\vec{w}))=\\
+=\Vert\vec{v}+\vec{w}\Vert^{2}+\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}+2(\vec{v}\cdot\vec{w}-(\vec{v}+\vec{w})\cdot\vec{v}-(\vec{v}+\vec{w})\cdot\vec{w})=\\
+=\Vert\vec{v}+\vec{w}\Vert^{2}+(\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}+2\vec{v}\cdot\vec{w})-2\Vert\vec{v}+\vec{w}\Vert^{2}=0
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\Vert\vec{v}+\vec{w}\Vert^{2}=\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}+2\vec{v}\cdot\vec{w}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\vec{v}\cdot\vec{w}=\frac{1}{2}(\Vert\vec{v}+\vec{w}\Vert^{2}-\Vert\vec{v}\Vert^{2}-\Vert\vec{w}\Vert^{2})$
+\end_inset
+
+ y por tanto si una aplicación lineal conserva normas también conserva productos
+ escalares.
+\end_layout
+
+\begin_layout Standard
+Propiedades de las transformaciones ortogonales:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $U\bot W\implies f(U)\bot f(W)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Su composición es ortogonal.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\Vert g(f(\vec{v}))\Vert=\Vert f(\vec{v})\Vert=\Vert\vec{v}\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Son inyectivas.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $(f(\vec{v})=\vec{0}\implies\Vert\vec{v}\Vert=\Vert f(\vec{v})\Vert=0\implies\vec{v}=\vec{0})\implies\text{Nuc}(f)=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+La inversa de una transformación ortogonal biyectiva es ortogonal.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\Vert f^{-1}(\vec{v})\Vert=\Vert f(f^{-1}(\vec{v}))\Vert=\Vert\vec{v}\Vert$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $V$
+\end_inset
+
+ tiene dimensión finita, sus transformaciones ortogonales son biyectivas
+ y 
+\begin_inset Formula ${\cal O}(V)$
+\end_inset
+
+ con la composición de aplicaciones es un grupo.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $V$
+\end_inset
+
+ un espacio vectorial de dimensión finita y 
+\begin_inset Formula ${\cal B}=\{\vec{v}_{1},\dots,\vec{v}_{n}\}$
+\end_inset
+
+ una base ortonormal de 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Otra base 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+ de 
+\begin_inset Formula $V$
+\end_inset
+
+ es ortonormal si 
+\begin_inset Formula $M_{{\cal B}{\cal B}'}$
+\end_inset
+
+ es ortogonal.
+ 
+\begin_inset Formula $f:V\rightarrow V$
+\end_inset
+
+ es ortogonal si y sólo si 
+\begin_inset Formula $M_{{\cal B}}(f)$
+\end_inset
+
+ es ortogonal.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $A=M_{{\cal B}}(f)$
+\end_inset
+
+, si 
+\begin_inset Formula $f$
+\end_inset
+
+ es ortogonal, 
+\begin_inset Formula $A^{t}\cdot A=(f(\vec{v}_{i})\cdot f(\vec{v}_{j}))_{ij}=(\vec{v}_{i}\cdot\vec{v}_{j})_{ij}=(\delta_{ij})_{ij}=I_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $A^{t}\cdot A=(f(\vec{v}_{i})\cdot f(\vec{v}_{j}))_{ij}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $A$
+\end_inset
+
+ es ortogonal, 
+\begin_inset Formula $f(\vec{v}_{i})\cdot f(\vec{v}_{j})=\delta_{ij}$
+\end_inset
+
+, por lo que si 
+\begin_inset Formula $\vec{v}=\sum_{i}r_{i}\vec{v}_{i}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f(\vec{v})=\sum_{i}r_{i}f(\vec{v}_{i})$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\Vert f(\vec{v})\Vert^{2}=(\sum_{i}r_{i}f(\vec{v}_{i}))(\sum_{j}r_{j}f(\vec{v}_{j}))=\sum_{i}\sum_{j}r_{i}r_{j}f(\vec{v}_{i})\cdot f(\vec{v}_{j})=\sum_{i}\sum_{j}r_{i}r_{j}\delta_{ij}=\sum_{i}r_{i}^{2}=\Vert\vec{v}\Vert^{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El determinante de una transformación ortogonal solo puede ser 
+\begin_inset Formula $1$
+\end_inset
+
+ o 
+\begin_inset Formula $-1$
+\end_inset
+
+, pues 
+\begin_inset Formula $1=\det(I_{n})=\det(A^{t})\det(A)=\det(A)^{2}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f\in{\cal O}(V)$
+\end_inset
+
+ es 
+\series bold
+positiva
+\series default
+ o 
+\series bold
+directa
+\series default
+ (
+\begin_inset Formula $f\in{\cal O}^{+}(V)$
+\end_inset
+
+) si 
+\begin_inset Formula $\det(f)=1$
+\end_inset
+
+, y es 
+\series bold
+negativa
+\series default
+ o 
+\series bold
+inversa
+\series default
+ (
+\begin_inset Formula $f\in{\cal O}^{-}(V)$
+\end_inset
+
+) si 
+\begin_inset Formula $\det(f)=-1$
+\end_inset
+
+.
+ Claramente 
+\begin_inset Formula ${\cal O}(V)={\cal O}^{+}(V)\dot{\cup}{\cal O}^{-}(V)$
+\end_inset
+
+.
+ Se cumple la 
+\series bold
+regla de los signos
+\series default
+: La composición de transformaciones del mismo signo es positiva, y la de
+ transformaciones de distinto signo es negativa.
+\end_layout
+
+\begin_layout Standard
+Los únicos valores propios que puede tener 
+\begin_inset Formula $f\in{\cal O}(V)$
+\end_inset
+
+ son 
+\begin_inset Formula $\pm1$
+\end_inset
+
+, y los subespacios 
+\begin_inset Formula $\text{Inv}(f)$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Opp}(f)$
+\end_inset
+
+, que pueden ser nulos, son ortogonales.
+ Además, si 
+\begin_inset Formula $\dim(V)$
+\end_inset
+
+ es impar, al menos uno de estos subespacios es no nulo.
+ 
+\series bold
+Demostración:
+\series default
+ El polinomio característico de 
+\begin_inset Formula $f$
+\end_inset
+
+ tiene pues grado impar y por tanto al menos una raíz real, que por lo anterior
+ debe ser 
+\begin_inset Formula $\pm1$
+\end_inset
+
+, y el correspondiente subespacio propio es no nulo.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $U$
+\end_inset
+
+ es un subespacio invariante de 
+\begin_inset Formula $f\in{\cal O}(V)$
+\end_inset
+
+, también lo es 
+\begin_inset Formula $U^{\bot}$
+\end_inset
+
+, y de hecho, 
+\begin_inset Formula $f(U)=U$
+\end_inset
+
+, 
+\begin_inset Formula $f(U^{\bot})=U^{\bot}$
+\end_inset
+
+, 
+\begin_inset Formula $f|_{U}\in{\cal O}(U)$
+\end_inset
+
+ y 
+\begin_inset Formula $f|_{U^{\bot}}\in{\cal O}(U^{\bot})$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Como 
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectiva y la dimensión finita, 
+\begin_inset Formula $f(U)\subseteq U$
+\end_inset
+
+ implica 
+\begin_inset Formula $f(U)=U$
+\end_inset
+
+, y por la conservación del producto escalar, 
+\begin_inset Formula $f(U^{\bot})\bot f(U)$
+\end_inset
+
+, luego 
+\begin_inset Formula $f(U^{\bot})\subseteq U^{\bot}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f(U^{\bot})=U^{\bot}$
+\end_inset
+
+.
+ Como 
+\begin_inset Formula $f$
+\end_inset
+
+ conserva el producto escalar, también lo conservan 
+\begin_inset Formula $f|_{U}$
+\end_inset
+
+ y 
+\begin_inset Formula $f|_{U^{\bot}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dadas 
+\begin_inset Formula $g\in{\cal O}(U)$
+\end_inset
+
+ y 
+\begin_inset Formula $h\in{\cal O}(U^{\bot})$
+\end_inset
+
+, existe una única 
+\begin_inset Formula $f\in{\cal O}(V)$
+\end_inset
+
+ con 
+\begin_inset Formula $f|_{U}=g$
+\end_inset
+
+ y 
+\begin_inset Formula $f|_{U^{\bot}}=h$
+\end_inset
+
+.
+ Se cumple entonces que si 
+\begin_inset Formula ${\cal B}_{1}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}_{2}$
+\end_inset
+
+ son bases ortonormales respectivas de 
+\begin_inset Formula $U$
+\end_inset
+
+ y 
+\begin_inset Formula $U^{\bot}$
+\end_inset
+
+ entonces
+\begin_inset Formula 
+\[
+M_{{\cal B}}(f)=\left(\begin{array}{c|c}
+M_{{\cal B}_{1}}(g) & 0\\
+\hline 0 & M_{{\cal B}_{2}}(h)
+\end{array}\right)
+\]
+
+\end_inset
+
+ 
+\series bold
+Demostración:
+\series default
+ Si 
+\begin_inset Formula $V=U\oplus W$
+\end_inset
+
+ y tenemos 
+\begin_inset Formula $g:U\rightarrow U$
+\end_inset
+
+ y 
+\begin_inset Formula $h:W\rightarrow W$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f:V\rightarrow V$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\vec{v}=\vec{u}+\vec{w}\mapsto g(\vec{u})+h(\vec{w})$
+\end_inset
+
+ es lineal y el único endomorfismo con 
+\begin_inset Formula $f|_{U}=g$
+\end_inset
+
+ y 
+\begin_inset Formula $f|_{W}=h$
+\end_inset
+
+.
+ Si además 
+\begin_inset Formula $W=U^{\bot}$
+\end_inset
+
+, y 
+\begin_inset Formula $g$
+\end_inset
+
+ y 
+\begin_inset Formula $h$
+\end_inset
+
+ son ortogonales, entonces por el teorema de Pitágoras, 
+\begin_inset Formula $\Vert f(\vec{u}+\vec{w})\Vert^{2}=\Vert g(\vec{u})+h(\vec{w})\Vert^{2}=\Vert g(\vec{u})\Vert^{2}+\Vert h(\vec{w})\Vert^{2}=\Vert\vec{u}\Vert^{2}+\Vert\vec{w}\Vert^{2}=\Vert\vec{u}+\vec{w}\Vert^{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+En adelante llamamos 
+\begin_inset Formula ${\cal E}_{n}$
+\end_inset
+
+ a cualquier espacio vectorial euclídeo isomorfo a 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ con el producto escalar ordinario, pues todos los de igual dimensión sobre
+ el mismo cuerpo son isomorfos.
+\end_layout
+
+\begin_layout Standard
+Dos bases 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal B}'$
+\end_inset
+
+ en 
+\begin_inset Formula ${\cal E}_{n}$
+\end_inset
+
+ son 
+\series bold
+equivalentes
+\series default
+ si 
+\begin_inset Formula $\det(M_{{\cal B}{\cal B}'})>0$
+\end_inset
+
+.
+ 
+\series bold
+Orientar
+\series default
+ el espacio 
+\begin_inset Formula ${\cal E}_{n}$
+\end_inset
+
+ es elegir en él una base, de modo que las bases equivalentes a esta son
+ 
+\series bold
+positivas
+\series default
+ o 
+\series bold
+directas
+\series default
+ y el resto son 
+\series bold
+negativas
+\series default
+ o 
+\series bold
+inversas
+\series default
+.
+\end_layout
+
+\begin_layout Section
+Transformaciones ortogonales en 
+\begin_inset Formula ${\cal E}_{1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Un vector en 
+\begin_inset Formula ${\cal E}_{1}$
+\end_inset
+
+ solo puede ser llevado por una transformación ortogonal a sí mismo y su
+ inverso, luego 
+\begin_inset Formula ${\cal O}^{+}({\cal E}_{1})=\{id_{{\cal E}_{1}}\}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal O}^{-}({\cal E}_{1})=\{-id_{{\cal E}_{1}}\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Tranformaciones ortogonales en 
+\begin_inset Formula ${\cal E}_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $M=M_{{\cal B}}(f)$
+\end_inset
+
+ para una base 
+\begin_inset Formula ${\cal B}$
+\end_inset
+
+ arbitraria.
+ Si 
+\begin_inset Formula $M=\left(\begin{array}{cc}
+a & b\\
+c & d
+\end{array}\right)$
+\end_inset
+
+ es ortogonal positiva, entonces 
+\begin_inset Formula $M^{-1}=M^{t}=\left(\begin{array}{cc}
+d & -c\\
+-b & a
+\end{array}\right)$
+\end_inset
+
+, luego 
+\begin_inset Formula $d=a$
+\end_inset
+
+ y 
+\begin_inset Formula $c=-b$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula 
+\[
+M=\left(\begin{array}{cc}
+a & -b\\
+b & a
+\end{array}\right)
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $a^{2}+b^{2}=1$
+\end_inset
+
+.
+ Escribimos 
+\begin_inset Formula ${\cal O}^{+}(2,\mathbb{R}):={\cal O}^{+}({\cal E}_{2})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $b=0$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $a^{2}=1$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $a=\pm1$
+\end_inset
+
+ y se obtienen las transformaciones 
+\begin_inset Formula $\pm id_{{\cal E}_{2}}$
+\end_inset
+
+.
+ En particular, 
+\begin_inset Formula $id_{{\cal E}_{2}}$
+\end_inset
+
+ cumple 
+\begin_inset Formula $\dim(\text{Inv}(f))=2$
+\end_inset
+
+ y 
+\begin_inset Formula $\dim(\text{Opp}(f))=0$
+\end_inset
+
+, mientras que 
+\begin_inset Formula $-id_{{\cal E}_{2}}$
+\end_inset
+
+ cumple lo contrario.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $b\neq0$
+\end_inset
+
+, el polinomio característico tiene raíces complejas, luego
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $\dim(\text{Inv}(f))=\dim(\text{Opp}(f))=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dadas 
+\begin_inset Formula $f,g\in{\cal O}^{+}({\cal E}_{2})$
+\end_inset
+
+, 
+\begin_inset Formula $f\circ g=g\circ f$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula 
+\[
+\left(\begin{array}{cc}
+a & -b\\
+b & a
+\end{array}\right)\left(\begin{array}{cc}
+c & -d\\
+d & c
+\end{array}\right)=\left(\begin{array}{cc}
+ac-bd & -ad-bc\\
+ad+bc & ac-bd
+\end{array}\right)=\left(\begin{array}{cc}
+c & -d\\
+d & c
+\end{array}\right)\left(\begin{array}{cc}
+a & -b\\
+b & a
+\end{array}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos a la aplicación 
+\begin_inset Formula $g_{\theta}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $M_{{\cal B}}(g_{\theta})=\left(\begin{array}{cc}
+\cos\theta & -\sin\theta\\
+\sin\theta & \cos\theta
+\end{array}\right)$
+\end_inset
+
+ la 
+\series bold
+rotación
+\series default
+ o 
+\series bold
+giro
+\series default
+ de ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+.
+ Se cumple que 
+\begin_inset Formula $g_{\theta'}\circ g_{\theta}=g_{\theta+\theta'}$
+\end_inset
+
+ y 
+\begin_inset Formula $g_{\theta}^{-1}=g_{-\theta}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $M=\left(\begin{array}{cc}
+a & b\\
+c & d
+\end{array}\right)$
+\end_inset
+
+ es ortogonal negativa, entonces 
+\begin_inset Formula $M^{-1}=M^{t}=\left(\begin{array}{cc}
+-d & c\\
+b & -a
+\end{array}\right)$
+\end_inset
+
+, luego 
+\begin_inset Formula $a=-d$
+\end_inset
+
+ y 
+\begin_inset Formula $b=c$
+\end_inset
+
+.
+ Por tanto
+\begin_inset Formula 
+\[
+M=\left(\begin{array}{cc}
+a & b\\
+b & -a
+\end{array}\right)
+\]
+
+\end_inset
+
+con 
+\begin_inset Formula $a^{2}+b^{2}=1$
+\end_inset
+
+.
+ Por el polinomio característico hallamos que 
+\begin_inset Formula $\text{Inv}(f)$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Opp}(f)$
+\end_inset
+
+ son rectas ortogonales, y decimos que 
+\begin_inset Formula $f$
+\end_inset
+
+ es la 
+\series bold
+simetría axial
+\series default
+ sobre 
+\begin_inset Formula $\text{Inv}(f)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Toda rotación puede expresarse como composición de 2 simetrías axiales,
+ y una de ellas puede elegirse arbitrariamente.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $f$
+\end_inset
+
+ la rotación y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ una simetría axial, entonces 
+\begin_inset Formula $\sigma':=\sigma\circ f$
+\end_inset
+
+ es negativa y por tanto una simetría axial.
+ Entonces 
+\begin_inset Formula $\sigma\circ\sigma'=\sigma\circ\sigma\circ f=f$
+\end_inset
+
+.
+ Si queremos que 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ aparezca a la derecha, hacemos un razonamiento análogo con 
+\begin_inset Formula $\sigma'':=f\circ\sigma$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Transformaciones ortogonales en 
+\begin_inset Formula ${\cal E}_{3}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $f\in{\cal O}({\cal E}_{3})$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $\dim(\text{Inv}(f))=3$
+\end_inset
+
+, todo vector de 
+\begin_inset Formula $V$
+\end_inset
+
+ es invariante y por tanto 
+\begin_inset Formula $f=id_{{\cal E}_{3}}$
+\end_inset
+
+, una transformación positiva.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\dim(\text{Inv}(f))=2$
+\end_inset
+
+, sea 
+\begin_inset Formula $H=\text{Inv}(f)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f|_{H^{\bot}}$
+\end_inset
+
+ es una transformación ortogonal de la recta 
+\begin_inset Formula $H^{\bot}$
+\end_inset
+
+ que no puede tener invariantes, luego 
+\begin_inset Formula $H^{\bot}=\text{Opp}(f)$
+\end_inset
+
+ y 
+\begin_inset Formula $\dim(\text{Opp}(f))=1$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f=\sigma_{H}$
+\end_inset
+
+ es la 
+\series bold
+simetría especular
+\series default
+ sobre 
+\begin_inset Formula $H$
+\end_inset
+
+, una transformación negativa.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\dim(\text{Inv}(f))=1$
+\end_inset
+
+, sea 
+\begin_inset Formula $\ell=\text{Inv}(f)$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f|_{\ell^{\bot}}$
+\end_inset
+
+ es una transformación ortogonal del plano 
+\begin_inset Formula $\ell^{\bot}$
+\end_inset
+
+ sin vectores invariantes, luego es una rotación distinta de la identidad,
+ de ángulo 
+\begin_inset Formula $\theta\neq0$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es la 
+\series bold
+rotación
+\series default
+ de eje 
+\begin_inset Formula $\ell$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, una transformación positiva.
+ En particular, si 
+\begin_inset Formula $\theta=\pi$
+\end_inset
+
+ (
+\series bold
+simetría axial
+\series default
+), entonces 
+\begin_inset Formula $f|_{\ell^{\bot}}=-id_{\ell^{\bot}}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\dim(\text{Opp}(f))=2$
+\end_inset
+
+, mientras que en otro caso 
+\begin_inset Formula $\dim(\text{Opp}(f))=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\dim(\text{Inv}(f))=0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\text{Opp}(f)\neq0$
+\end_inset
+
+.
+ Sea entonces 
+\begin_inset Formula $\vec{v}\in\text{Opp}(f)$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\ell:=<\vec{v}>\subseteq\text{Opp}(f)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $f|_{\ell}=-id_{\ell}$
+\end_inset
+
+, mientras que 
+\begin_inset Formula $f|_{\ell^{\bot}}$
+\end_inset
+
+ es una transformación ortogonal del plano 
+\begin_inset Formula $\ell^{\bot}$
+\end_inset
+
+ sin vectores invariantes y por tanto una rotación distinta de la identidad,
+ de ángulo 
+\begin_inset Formula $\theta\neq0$
+\end_inset
+
+.
+ Decimos que 
+\begin_inset Formula $f$
+\end_inset
+
+ es una 
+\series bold
+rotación con simetría
+\series default
+ de eje 
+\begin_inset Formula $\ell$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, una transformación negativa.
+ En particular, si 
+\begin_inset Formula $\theta=\pi$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f|_{\ell^{\bot}}=-id_{\ell^{\bot}}$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f=-id_{{\cal E}_{3}}$
+\end_inset
+
+, con 
+\begin_inset Formula $\dim(\text{Opp}(f))=3$
+\end_inset
+
+, mientras que si 
+\begin_inset Formula $\theta\neq\pi$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\text{Opp}(f|_{\ell^{\bot}})=0$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\dim(\text{Opp}(f))=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Así pues, en general, 
+\begin_inset Formula ${\cal O}^{+}({\cal E}_{3})$
+\end_inset
+
+ son las rotaciones (incluyendo de ángulo 0) y 
+\begin_inset Formula ${\cal O}^{-}({\cal E}_{3})$
+\end_inset
+
+ son las rotaciones con simetría.
+\end_layout
+
+\begin_layout Standard
+Para construir la matriz de una transformación en 
+\begin_inset Formula ${\cal E}_{3}$
+\end_inset
+
+, tomamos una base 
+\begin_inset Quotes cld
+\end_inset
+
+cómoda
+\begin_inset Quotes crd
+\end_inset
+
+ 
+\begin_inset Formula ${\cal B}=\{\vec{v}_{1},\vec{v}_{2},\vec{v}_{3}\}$
+\end_inset
+
+ y aplicamos la fórmula de cambio de base.
+ Entonces:
+\end_layout
+
+\begin_layout Standard
+\align center
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="3">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<row>
+<cell alignment="center" valignment="top" bottomline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rotación (eje 
+\begin_inset Formula $<\vec{v}_{1}>$
+\end_inset
+
+, ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+)
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rotación con simetría (ídem)
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Matriz
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\left(\begin{array}{ccc}
+1 & 0 & 0\\
+0 & \cos\theta & -\sin\theta\\
+0 & \sin\theta & \cos\theta
+\end{array}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\left(\begin{array}{ccc}
+-1 & 0 & 0\\
+0 & \cos\theta & -\sin\theta\\
+0 & \sin\theta & \cos\theta
+\end{array}\right)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Traza
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $1+2\cos\theta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-1+2\cos\theta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Det.
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $-1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Aquí se incluyen la identidad, menos identidad y simetrías axiales y especulares
+ como casos especiales de estos dos.
+ La traza de un endomorfismo (suma de los elementos de la diagonal de la
+ matriz) no depende de la base, pues 
+\begin_inset Formula $\text{tr}(M')=\text{tr}(P^{-1}MP)=\text{tr}(MPP^{-1})=\text{tr}(M)$
+\end_inset
+
+, pudiendo servir para determinar el ángulo de una transformación dada su
+ matriz en cualquier base.
+\end_layout
+
+\begin_layout Standard
+Toda rotación se expresa como composición de 2 simetrías especulares, de
+ las que una se puede elegir arbitrariamente siempre que su base contenga
+ al eje de la rotación.
+ Por tanto toda rotación con simetría se expresa como composición de tres
+ simetrías especulares.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $f$
+\end_inset
+
+ una rotación de eje 
+\begin_inset Formula $F$
+\end_inset
+
+ y 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ la simetría especular sobre un plano que contiene a 
+\begin_inset Formula $F$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\sigma':=\sigma\circ f$
+\end_inset
+
+ es negativa con vectores invariantes y por tanto otra simetría especular,
+ y entonces 
+\begin_inset Formula $\sigma\circ\sigma'=\sigma\circ\sigma\circ f=f$
+\end_inset
+
+.
+ Si queremos que 
+\begin_inset Formula $\sigma$
+\end_inset
+
+ aparezca a la derecha basta hacer lo mismo con 
+\begin_inset Formula $\sigma'':=f\circ\sigma$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/gae/n5.lyx b/gae/n5.lyx
new file mode 100644
index 0000000..8332603
--- /dev/null
+++ b/gae/n5.lyx
@@ -0,0 +1,887 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style swiss
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+Una 
+\series bold
+isometría
+\series default
+ o 
+\series bold
+movimiento
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+ es una aplicación 
+\begin_inset Formula $f:E\rightarrow E$
+\end_inset
+
+ con
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $d(P,Q)=d(f(P),f(Q))$
+\end_inset
+
+ (también se puede hablar de isometrías entre espacios distintos).
+ El conjunto que forman es el 
+\series bold
+grupo de los movimientos
+\series default
+ de 
+\begin_inset Formula $E$
+\end_inset
+
+, escrito 
+\begin_inset Formula $\text{Is}(E)$
+\end_inset
+
+.
+ Una aplicación 
+\begin_inset Formula $f:E\rightarrow E$
+\end_inset
+
+ es un movimiento si y sólo si es afín y 
+\begin_inset Formula $\overrightarrow{f}:V\rightarrow V$
+\end_inset
+
+ es ortogonal.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Fijado 
+\begin_inset Formula $A\in E$
+\end_inset
+
+, demostramos que si 
+\begin_inset Formula $\ell:V\rightarrow V$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\ell(\vec{v}):=\overrightarrow{f(A)f(A+\vec{v})}$
+\end_inset
+
+ es lineal, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es afín con 
+\begin_inset Formula $\overrightarrow{f}=\ell$
+\end_inset
+
+.
+ En efecto, para 
+\begin_inset Formula $P\in E$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $\ell(\overrightarrow{AP})=\overrightarrow{f(A)f(A+\overrightarrow{AP})}=\overrightarrow{f(A)f(P)}$
+\end_inset
+
+, y dados 
+\begin_inset Formula $P,Q\in E$
+\end_inset
+
+, 
+\begin_inset Formula $\ell(\overrightarrow{PQ})=\ell(-\overrightarrow{AP}+\overrightarrow{AQ})=-\ell(\overrightarrow{AP})+\ell(\overrightarrow{AQ})=-\overrightarrow{f(A)f(P)}+\overrightarrow{f(A)f(Q)}=\overrightarrow{f(P)f(Q)}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+A continuación veamos que 
+\begin_inset Formula $\ell$
+\end_inset
+
+ es ortogonal, y por tanto será lineal y 
+\begin_inset Formula $f$
+\end_inset
+
+ será afín con 
+\begin_inset Formula $\overrightarrow{f}=\ell$
+\end_inset
+
+.
+ Dados 
+\begin_inset Formula $\vec{v},\vec{w}\in V$
+\end_inset
+
+, si 
+\begin_inset Formula $P:=A+\vec{v}$
+\end_inset
+
+ y 
+\begin_inset Formula $Q:=A+\vec{w}$
+\end_inset
+
+, deducimos 
+\begin_inset Formula $\vec{v}\cdot\vec{w}=\frac{1}{2}\left(\Vert\vec{v}\Vert^{2}+\Vert\vec{w}\Vert^{2}-\Vert\vec{w}-\vec{v}\Vert^{2}\right)$
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Formula $=\frac{1}{2}\left(\Vert\overrightarrow{AP}\Vert^{2}+\Vert\overrightarrow{AQ}\Vert^{2}-\Vert\overrightarrow{PQ}\Vert^{2}\right)=\frac{1}{2}\left(d(A,P)^{2}+d(A,Q)^{2}-d(P,Q)^{2}\right)$
+\end_inset
+
+.
+ Pero del mismo modo, 
+\begin_inset Formula $\ell(\vec{v})\cdot\ell(\vec{w})=\frac{1}{2}\left(d(\ell(A),\ell(P))^{2}+d(\ell(A),\ell(Q))^{2}-d(\ell(P),\ell(Q))^{2}\right)$
+\end_inset
+
+, y como 
+\begin_inset Formula $f$
+\end_inset
+
+ conserva distancias, entonces 
+\begin_inset Formula $\ell(\vec{v})\cdot\ell(\vec{w})=\vec{v}\cdot\vec{w}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $d(P,Q)=\Vert\overrightarrow{PQ}\Vert=\Vert\overrightarrow{f}(\overrightarrow{PQ})\Vert=\Vert\overrightarrow{f(P)f(Q)}\Vert=d(f(P),f(Q))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Propiedades: Si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ son isometrías:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula ${\cal L}_{1}\bot{\cal L}_{2}\implies f({\cal L}_{1})\bot f({\cal L}_{2})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f\circ g$
+\end_inset
+
+ es una isometría.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es biyectiva, 
+\begin_inset Formula $f^{-1}$
+\end_inset
+
+ es una isometría.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $f$
+\end_inset
+
+ es inyectiva.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\dim(E)<\infty$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Is}(E)$
+\end_inset
+
+ es un grupo con la composición de aplicaciones.
+\end_layout
+
+\begin_layout Standard
+Un movimiento 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+positivo/directo
+\series default
+ o 
+\series bold
+negativo/inverso
+\series default
+ según lo sea 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+.
+ Llamamos 
+\begin_inset Formula $\text{Is}^{+}(E)$
+\end_inset
+
+ al conjunto de todos los movimientos positivos de 
+\begin_inset Formula $E$
+\end_inset
+
+, e 
+\begin_inset Formula $\text{Is}^{-}(E)$
+\end_inset
+
+ al de todos los negativos.
+\end_layout
+
+\begin_layout Section
+Movimientos en 
+\begin_inset Formula $E_{1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\overrightarrow{f}=id$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f=t_{\vec{v}}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}=\overrightarrow{Qf(Q)}$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+ arbitrario.
+ Si 
+\begin_inset Formula $\overrightarrow{f}=-id$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f=s_{P}$
+\end_inset
+
+ con 
+\begin_inset Formula $P=\frac{Q+f(Q)}{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+ arbitrario.
+\end_layout
+
+\begin_layout Section
+Movimientos en 
+\begin_inset Formula $E_{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Además de los dos casos posibles en 
+\begin_inset Formula $E_{1}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}$
+\end_inset
+
+ es una simetría ortogonal, si hay puntos fijos entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es la 
+\series bold
+simetría ortogonal (afín)
+\series default
+ de base 
+\begin_inset Formula $\text{Fix}(f)$
+\end_inset
+
+ (y con dirección 
+\begin_inset Formula $\text{dir}(\text{Fix}(f))^{\bot}$
+\end_inset
+
+), y de lo contrario es la 
+\series bold
+simetría ortogonal con deslizamiento
+\series default
+ de base 
+\begin_inset Formula ${\cal L}=A+\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+ y con vector de deslizamiento 
+\begin_inset Formula $\vec{v}=\overrightarrow{Af(A)}$
+\end_inset
+
+, siendo 
+\begin_inset Formula $A:=\frac{Q+f(Q)}{2}$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+ arbitrario, de modo que 
+\begin_inset Formula $f=s_{{\cal L}}\circ t_{\vec{v}}=t_{\vec{v}}\circ s_{{\cal L}}$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+En efecto, dado 
+\begin_inset Formula $Q\in E$
+\end_inset
+
+, si 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\vec{v}+\vec{w}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}\in W=\text{Inv}(\overrightarrow{f})$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}\in W^{\bot}$
+\end_inset
+
+ y llamamos 
+\begin_inset Formula $A:=\frac{Q+f(Q)}{2}=Q+\frac{1}{2}(\vec{v}+\vec{w})$
+\end_inset
+
+, como 
+\begin_inset Formula $\overrightarrow{f}=\sigma_{W}$
+\end_inset
+
+ es la simetría de base 
+\begin_inset Formula $W$
+\end_inset
+
+ y dirección 
+\begin_inset Formula $W^{\bot}$
+\end_inset
+
+, se tiene 
+\begin_inset Formula $\overrightarrow{f}(\overrightarrow{QA})=\overrightarrow{f}(\frac{1}{2}(\vec{v}+\vec{w}))=\frac{1}{2}\vec{v}-\frac{1}{2}\vec{w}$
+\end_inset
+
+, con lo que si 
+\begin_inset Formula $g=t_{-\vec{v}}\circ f$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $g(A)=(t_{-\vec{v}}\circ f)(A)=f(A)-\vec{v}=f(Q)+\overrightarrow{f}(\overrightarrow{QA})-\vec{v}=f(Q)-\frac{1}{2}\vec{v}-\frac{1}{2}\vec{w}-\vec{v}=f(Q)-\frac{1}{2}(\vec{v}+\vec{w})=A$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $\text{Fix}(g)\neq\emptyset$
+\end_inset
+
+ y como 
+\begin_inset Formula $\overrightarrow{g}=\overrightarrow{f}$
+\end_inset
+
+, resulta 
+\begin_inset Formula $g=s_{A+\text{Inv}(\overrightarrow{g})}=s_{{\cal L}}$
+\end_inset
+
+ y 
+\begin_inset Formula $f=t_{\vec{v}}\circ g$
+\end_inset
+
+, y es fácil comprobar que 
+\begin_inset Formula $t_{\vec{v}}\circ g=g\circ t_{\vec{v}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}=g_{\theta}$
+\end_inset
+
+ es la rotación de ángulo 
+\begin_inset Formula $\theta\neq0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f=\rho_{P,\theta}$
+\end_inset
+
+ es la 
+\series bold
+rotación
+\series default
+ de centro 
+\begin_inset Formula $P$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, siendo 
+\begin_inset Formula $P$
+\end_inset
+
+ el único punto fijo de 
+\begin_inset Formula $f$
+\end_inset
+
+, pues 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Movimientos en 
+\begin_inset Formula $E_{3}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lo dicho respecto a las traslaciones y simetrías también se aplica aquí,
+ pero también se pueden dar otros dos casos.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}$
+\end_inset
+
+ es la rotación de eje 
+\begin_inset Formula $F$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, si hay puntos fijos entonces 
+\begin_inset Formula $f=\rho_{\ell,\theta}$
+\end_inset
+
+ es la 
+\series bold
+rotación
+\series default
+ de eje 
+\begin_inset Formula $\ell=\text{Fix}(f)$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, y de lo contrario 
+\begin_inset Formula $f=t_{\vec{v}}\circ\rho_{\ell,\theta}=\rho_{\ell,\theta}\circ t_{\vec{v}}$
+\end_inset
+
+ es la 
+\series bold
+rotación con deslizamiento
+\series default
+ o 
+\series bold
+movimiento helicoidal
+\series default
+ de eje 
+\begin_inset Formula $\ell$
+\end_inset
+
+, ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+ y vector de deslizamiento 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+, donde 
+\begin_inset Formula $\vec{v}=\pi_{F}(\overrightarrow{Qf(Q)})$
+\end_inset
+
+ para 
+\begin_inset Formula $Q\in E_{3}$
+\end_inset
+
+ arbitrario y 
+\begin_inset Formula $\ell=\text{Fix}(t_{-\vec{v}}\circ f)$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Todo movimiento 
+\begin_inset Formula $f:E_{3}\rightarrow E_{3}$
+\end_inset
+
+ con 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}$
+\end_inset
+
+ para 
+\begin_inset Formula $\theta\neq0$
+\end_inset
+
+ y 
+\begin_inset Formula $\text{Fix}(f)=\emptyset$
+\end_inset
+
+ es un movimiento helicoidal con los elementos mencionados, y viceversa.
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $Q\in E_{3}$
+\end_inset
+
+ arbitrario y 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\vec{v}+\vec{w}$
+\end_inset
+
+ con 
+\begin_inset Formula $\vec{v}\in F$
+\end_inset
+
+ y 
+\begin_inset Formula $\vec{w}\in F^{\bot}$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ es la proyección ortogonal de 
+\begin_inset Formula $\overrightarrow{Qf(Q)}$
+\end_inset
+
+ sobre 
+\begin_inset Formula $F$
+\end_inset
+
+.
+ Sean ahora 
+\begin_inset Formula $g:=t_{-\vec{v}}\circ f$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal H}:=Q+F^{\bot}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $g({\cal H})\subseteq{\cal H}$
+\end_inset
+
+, pues 
+\begin_inset Formula $Q'\in{\cal H}\implies\exists\vec{x}\in F^{\bot}:Q'=Q+\vec{x}\implies g(Q')=g(Q+\vec{x})=f(Q+\vec{x})-\vec{v}=f(Q)-\vec{v}+\overrightarrow{f}(\vec{x})=Q+\vec{w}+\overrightarrow{f}(\vec{x})\in Q+F^{\bot}={\cal H}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $g|_{{\cal H}}$
+\end_inset
+
+ es un movimiento para el que 
+\begin_inset Formula $\overrightarrow{g}|_{F^{\bot}}=\overrightarrow{f}|_{F^{\bot}}$
+\end_inset
+
+ es una rotación, luego existe 
+\begin_inset Formula $P\in{\cal H}$
+\end_inset
+
+ con 
+\begin_inset Formula $g(P)=P$
+\end_inset
+
+.
+ Esto implica 
+\begin_inset Formula $\vec{v}\neq\vec{0}$
+\end_inset
+
+, pues de lo contrario sería 
+\begin_inset Formula $f=g$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ tendría puntos fijos.
+ Deducimos pues que 
+\begin_inset Formula $g$
+\end_inset
+
+ es la rotación 
+\begin_inset Formula $\rho_{\ell,\theta}$
+\end_inset
+
+ con 
+\begin_inset Formula $\ell=\text{Fix}(g)=\text{Fix}(t_{-\vec{v}}\circ f)$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f=t_{\vec{v}}\circ g$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $g:=\rho_{\ell,\theta}$
+\end_inset
+
+, para un 
+\begin_inset Formula $Q\in E_{3}$
+\end_inset
+
+ arbitrario, 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\overrightarrow{Q(g(Q)+\vec{v})}=\vec{v}+\overrightarrow{Qg(Q)}$
+\end_inset
+
+, donde 
+\begin_inset Formula $\vec{v}\in F$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{Qg(Q)}\in F^{\bot}$
+\end_inset
+
+, luego 
+\begin_inset Formula $\vec{v}$
+\end_inset
+
+ es la proyección ortogonal de 
+\begin_inset Formula $\overrightarrow{Qf(Q)}$
+\end_inset
+
+ sobre 
+\begin_inset Formula $F$
+\end_inset
+
+.
+ Esto prueba que 
+\begin_inset Formula $\text{Fix}(f)=\emptyset$
+\end_inset
+
+, pues de lo contrario se tendría 
+\begin_inset Formula $\overrightarrow{Qf(Q)}=\vec{0}$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\vec{v}=\vec{0}$
+\end_inset
+
+ y 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+Si 
+\begin_inset Formula $\overrightarrow{f}=\rho_{F,\theta}\circ\sigma_{F^{\bot}}$
+\end_inset
+
+ es una rotación con simetría, entonces 
+\begin_inset Formula $f=\rho_{\ell,\theta}\circ s_{{\cal H}}=s_{{\cal H}}\circ p_{\ell,\theta}$
+\end_inset
+
+ es una 
+\series bold
+rotación con simetría especular
+\series default
+ de base 
+\begin_inset Formula ${\cal H}$
+\end_inset
+
+ y ángulo 
+\begin_inset Formula $\theta$
+\end_inset
+
+, donde 
+\begin_inset Formula $\ell=P+F$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal H}=P+F^{\bot}$
+\end_inset
+
+ siendo 
+\begin_inset Formula $P$
+\end_inset
+
+ el único punto fijo de 
+\begin_inset Formula $f$
+\end_inset
+
+ (pues 
+\begin_inset Formula $\text{Inv}(\overrightarrow{f})=0$
+\end_inset
+
+).
+\end_layout
+
+\end_body
+\end_document