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Diffstat (limited to 'gae')
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| -rw-r--r-- | gae/n1.lyx | 1753 | ||||
| -rw-r--r-- | gae/n1b.lyx | 1010 | ||||
| -rw-r--r-- | gae/n2.lyx | 1978 | ||||
| -rw-r--r-- | gae/n3.lyx | 1832 | ||||
| -rw-r--r-- | gae/n4.lyx | 1509 | ||||
| -rw-r--r-- | gae/n5.lyx | 887 |
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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 +\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 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" +\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 +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 +\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 +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 |