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+#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