diff options
Diffstat (limited to 'ac/n3.lyx')
| -rw-r--r-- | ac/n3.lyx | 1442 |
1 files changed, 1439 insertions, 3 deletions
@@ -161,6 +161,55 @@ producto por escalares \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Equivalentemente, el producto currificado es un homomorfismo de anillos + +\begin_inset Formula $A\to\text{End}(M)$ +\end_inset + +, donde +\begin_inset Formula $\text{End}(M)$ +\end_inset + + es el anillo de los endomorfismos del grupo abeliano +\begin_inset Formula $M$ +\end_inset + + con la suma por componentes y la composición como producto. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Propiedades: \end_layout @@ -304,7 +353,7 @@ anulador \end_inset a -\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M\mid Xm=0\}\leq_{A}M$ +\begin_inset Formula $\text{ann}_{M}(X)\coloneqq\{m\in M\mid Xm=0\}$ \end_inset . @@ -665,6 +714,10 @@ Si \begin_inset Formula $\text{ann}_{M}(X)\leq_{A}M$ \end_inset +, y en particular +\begin_inset Formula $\text{ann}_{A}(X)\trianglelefteq A$ +\end_inset + . \end_layout @@ -676,6 +729,104 @@ Si \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +8. +\end_layout + +\end_inset + +Para +\begin_inset Formula $I\trianglelefteq A$ +\end_inset + + y +\begin_inset Formula $X\subseteq_{A}M$ +\end_inset + +, +\begin_inset Formula $IX\leq_{A}M$ +\end_inset + +, y en particular, para +\begin_inset Formula $m\in M$ +\end_inset + +, +\begin_inset Formula $Im=\{bm\}_{b\in I}\leq_{A}M$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +9. +\end_layout + +\end_inset + +Para +\begin_inset Formula $S\subseteq A$ +\end_inset + + y +\begin_inset Formula $N\leq_{A}M$ +\end_inset + +, +\begin_inset Formula $SN\leq_{A}M$ +\end_inset + +, y en particular, para +\begin_inset Formula $a\in A$ +\end_inset + +, +\begin_inset Formula $aN=\{an\}_{n\in N}\leq_{A}M$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Si \begin_inset Formula $N\leq_{A}M$ \end_inset @@ -1109,6 +1260,103 @@ Un . \end_layout +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Que dos submódulos de +\begin_inset Formula $_{A}M$ +\end_inset + + sean isomorfos no significa que lo sean los módulos cociente de +\begin_inset Formula $M$ +\end_inset + + entre ellos, ni al revés. + Por ejemplo, si +\begin_inset Formula $_{\mathbb{Z}}M\coloneqq\mathbb{Z}_{3}\oplus\mathbb{Z}_{9}$ +\end_inset + +, +\begin_inset Formula $K\coloneqq\mathbb{Z}_{3}\oplus0$ +\end_inset + +, +\begin_inset Formula $N\coloneqq0\oplus\mathbb{Z}_{9}$ +\end_inset + + y +\begin_inset Formula $L=((0,6))$ +\end_inset + +, +\begin_inset Formula $K\cong L$ +\end_inset + + pero +\begin_inset Formula $\frac{M}{K}\ncong\frac{M}{L}$ +\end_inset + +, y +\begin_inset Formula $\frac{M}{K+L}\cong\frac{M}{N}$ +\end_inset + + pero +\begin_inset Formula $K+L\ncong N$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Si +\begin_inset Formula $\phi:M\to M'$ +\end_inset + + es un +\begin_inset Formula $A$ +\end_inset + +-isomorfismo, para +\begin_inset Formula $N\leq_{A}M$ +\end_inset + +, +\begin_inset Formula $\frac{M}{N}\cong\frac{M'}{\phi(N)}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + \begin_layout Section Restricción de escalares \end_layout @@ -1616,6 +1864,94 @@ Finalmente, estas operaciones son inversas una de la otra, pues para \end_layout \end_deeper +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $V$ +\end_inset + + y +\begin_inset Formula $W$ +\end_inset + + +\begin_inset Formula $K$ +\end_inset + +-espacios vectoriales y +\begin_inset Formula $f:V\to V$ +\end_inset + + y +\begin_inset Formula $g:V\to V$ +\end_inset + + +\begin_inset Formula $K$ +\end_inset + +-endomorfismos, un +\begin_inset Formula $K[X]$ +\end_inset + +-homomorfismo entre los +\begin_inset Formula $K[X]$ +\end_inset + +-módulos asociados a +\begin_inset Formula $(V,f)$ +\end_inset + + y +\begin_inset Formula $(W,g)$ +\end_inset + + es precisamente una aplicación +\begin_inset Formula $K$ +\end_inset + +-lineal +\begin_inset Formula $\phi:V\to W$ +\end_inset + + con +\begin_inset Formula $\phi\circ f=g\circ\phi$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + \begin_layout Section Teoremas de isomorfía \end_layout @@ -1928,8 +2264,73 @@ Sea \end_layout \end_deeper +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Una +\series bold +clase de isomorfía +\series default + es una clase de equivalencia por la relación +\begin_inset Quotes cld +\end_inset + +ser isomorfos +\begin_inset Quotes crd +\end_inset + +. + Para +\begin_inset Formula $I,J\trianglelefteq A$ +\end_inset + +, si +\begin_inset Formula $\frac{A}{I}\cong\frac{A}{J}$ +\end_inset + + como +\begin_inset Formula $A$ +\end_inset + +-módulos entonces +\begin_inset Formula $I=J$ +\end_inset + +, pero esto no es válido si el isomorfismo es de anillos. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + \begin_layout Section -Operaciones con submódulos +Sistemas generadores \end_layout \begin_layout Standard @@ -2291,6 +2692,265 @@ Si \end_deeper \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +9. +\end_layout + +\end_inset + +Si +\begin_inset Formula $N\leq_{A}M$ +\end_inset + + y +\begin_inset Formula $\frac{M}{N}$ +\end_inset + + son finitamente generados, +\begin_inset Formula $M$ +\end_inset + + es finitamente generado. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +10. +\end_layout + +\end_inset + +Si +\begin_inset Formula $N,K\leq_{A}M$ +\end_inset + +, +\begin_inset Formula $N\cap K\eqqcolon(x_{1},\dots,x_{r})$ +\end_inset + +, +\begin_inset Formula $N+K\eqqcolon(y_{1},\dots,y_{s})$ +\end_inset + + y, para +\begin_inset Formula $j\in\{1,\dots,s\}$ +\end_inset + +, +\begin_inset Formula $y_{j}\eqqcolon n_{j}+k_{j}$ +\end_inset + + con +\begin_inset Formula $n_{j}\in N$ +\end_inset + + y +\begin_inset Formula $k_{j}\in K$ +\end_inset + +, entonces +\begin_inset Formula $N=(x_{1},\dots,x_{r},n_{1},\dots,n_{s})$ +\end_inset + + y +\begin_inset Formula $K=(x_{1},\dots,x_{r},k_{1},\dots,k_{s})$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +11. +\end_layout + +\end_inset + +Dado un entero +\begin_inset Formula $q\geq2$ +\end_inset + +, +\begin_inset Formula $\mathbb{Z}\left[\frac{1}{q}\right]=\left\{ \frac{a}{q^{n}}\right\} _{a\in\mathbb{Z},n\in\mathbb{N}}\leq_{\mathbb{Z}}\mathbb{Q}$ +\end_inset + + no es finitamente generado. + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +12. +\end_layout + +\end_inset + +Los epimorfismos conservan los conjuntos generadores. +\end_layout + +\begin_layout Standard + +\series bold +Lema de Nakayama: +\series default + Dados +\begin_inset Formula $_{A}M$ +\end_inset + + y +\begin_inset Formula $J\leq A$ +\end_inset + + con +\begin_inset Formula $J\subseteq\text{Jac}A$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $M$ +\end_inset + + es finitamente generado y +\begin_inset Formula $JM=M$ +\end_inset + + entonces +\begin_inset Formula $M=0$ +\end_inset + +. + Esto no se cumple si +\begin_inset Formula $_{A}M$ +\end_inset + + no es finitamente generado, pues por ejemplo +\begin_inset Formula $\mathbb{Q}$ +\end_inset + + visto como +\begin_inset Formula $\mathbb{Z}_{(p)}$ +\end_inset + +-módulo cumple +\begin_inset Formula $\text{Jac}(\mathbb{Z}_{p}(\mathbb{Q}))=\mathbb{Q}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $M$ +\end_inset + + es finitamente generado, el único +\begin_inset Formula $N\leq_{A}M$ +\end_inset + + con +\begin_inset Formula $M=JM+N$ +\end_inset + + es +\begin_inset Formula $M$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $(A,J,K)$ +\end_inset + + es un anillo local, +\begin_inset Formula $\frac{M}{JM}$ +\end_inset + + es anulado por +\begin_inset Formula $J$ +\end_inset + + ( +\begin_inset Formula $J\subseteq\text{ann}_{A}(\frac{M}{JM})$ +\end_inset + +), luego es un +\begin_inset Formula $K$ +\end_inset + +-espacio vectorial. + Si además +\begin_inset Formula $M$ +\end_inset + + es finitamente generado, +\begin_inset Formula $\frac{M}{JM}$ +\end_inset + + es de dimensión finita, y si +\begin_inset Formula $_{K}\frac{M}{JM}=(\overline{m_{1}},\dots,\overline{m_{n}})$ +\end_inset + + entonces +\begin_inset Formula $_{A}M=(m_{1},\dots,m_{n})$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Section +Sumas directas +\end_layout + +\begin_layout Standard Sean \begin_inset Formula $\{N_{i}\}_{i\in I}\subseteq{\cal L}(_{A}M)$ \end_inset @@ -2571,6 +3231,100 @@ La unión de un conjunto generador de \end_deeper \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + +Si +\begin_inset Formula $J\trianglelefteq A$ +\end_inset + + y +\begin_inset Formula $_{A}M=\bigoplus_{i\in I}M_{i}$ +\end_inset + +: +\end_layout + +\begin_layout Enumerate +Dado un +\begin_inset Formula $A$ +\end_inset + +-isomorfismo +\begin_inset Formula $\phi:M\to N$ +\end_inset + +, +\begin_inset Formula $N=\bigoplus_{i\in I}f(M_{i})$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\text{ann}_{M}(J)=\bigoplus_{i\in I}\text{ann}_{M_{i}}(J)$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Formula $\text{ann}_{A}(M)=\bigcap_{i\in I}\text{ann}_{A}(M_{i})$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +Si +\begin_inset Formula $A$ +\end_inset + + es un DIP, +\begin_inset Formula $I$ +\end_inset + + es finito y +\begin_inset Formula $\text{ann}_{A}(M_{i})=(b_{i})$ +\end_inset + + para cada +\begin_inset Formula $i\in I$ +\end_inset + +, entonces +\begin_inset Formula $\text{ann}_{A}(M)=(\text{lcm}_{i\in I}b_{i})$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard \begin_inset Formula $N\leq_{A}M$ \end_inset @@ -3282,6 +4036,88 @@ TODO ejercicio Saorín 2 \end_layout +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +8. +\end_layout + +\end_inset + +Si +\begin_inset Formula $e\in A$ +\end_inset + + es idempotente, +\begin_inset Formula $eM$ +\end_inset + + es sumando directo de +\begin_inset Formula $M$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +9. +\end_layout + +\end_inset + +Si +\begin_inset Formula $f:M\to M$ +\end_inset + + es un +\begin_inset Formula $A$ +\end_inset + +-endomorfismo idempotente, +\begin_inset Formula $M=\ker f\oplus\text{Im}f$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + \begin_layout Section Módulos libres \end_layout @@ -3600,7 +4436,78 @@ begin{exinfo} \end_inset + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +8. +\end_layout + +\end_inset + +Los epimorfismos conservan la independencia lineal. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +9. +\end_layout + +\end_inset + Los isomorfismos conservan bases. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +10. +\end_layout + +\end_inset + +Un +\begin_inset Formula $\mathbb{Z}$ +\end_inset + +-submódulo de +\begin_inset Formula $\mathbb{Q}$ +\end_inset + + es libre si y sólo si es cíclico, si y solo si es finitamente generado. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +11. +\end_layout + +\end_inset + +Un anillo +\begin_inset Formula $A$ +\end_inset + + es un cuerpo si y sólo si todo +\begin_inset Formula $A$ +\end_inset + +-módulo es libre. +\end_layout + +\begin_layout Standard \begin_inset ERT status open @@ -4209,6 +5116,43 @@ Sean \end_layout \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\begin_inset Formula $_{A}N\in{\cal L}(_{A}M)$ +\end_inset + + es +\series bold +finitamente cogenerado +\series default + si es cocompacto. +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard \begin_inset Formula $_{A}M$ \end_inset @@ -4497,6 +5441,105 @@ Como todos sus subgrupos son los de esta cadena, \end_deeper \begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +5. +\end_layout + +\end_inset + + +\begin_inset Formula $\frac{\mathbb{Q}}{\mathbb{Z}}=\bigoplus_{p}\mathbb{Z}_{p^{\infty}}$ +\end_inset + +. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +6. +\end_layout + +\end_inset + +Si +\begin_inset Formula $_{A}M$ +\end_inset + + es noetheriano, todo +\begin_inset Formula $A$ +\end_inset + +-endomorfismo suprayectivo en +\begin_inset Formula $M$ +\end_inset + + es inyectivo. +\end_layout + +\begin_layout Enumerate +\begin_inset Argument item:1 +status open + +\begin_layout Plain Layout +7. +\end_layout + +\end_inset + +Si +\begin_inset Formula $_{A}M$ +\end_inset + + es artiniano, todo +\begin_inset Formula $A$ +\end_inset + +-endomorfismo inyectivo en +\begin_inset Formula $M$ +\end_inset + + es suprayectivo. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{exinfo} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard Una \series bold sucesión exacta corta @@ -5032,7 +6075,7 @@ Si \end_inset -módulo, y en particular -\begin_inset Formula ${\cal L}(_{A}M)\cong{\cal L}(_{A_{1}}M_{1})\times\dots\times{\cal L}(_{A_{n}}M_{n})$ +\begin_inset Formula ${\cal L}(_{A}M)\cong\prod_{i=1}^{m}{\cal L}(_{A_{i}}M_{i})$ \end_inset . @@ -5306,5 +6349,398 @@ de longitud finita -módulo finitamente generado es de longitud finita. \end_layout +\begin_layout Section +Módulos y matrices +\end_layout + +\begin_layout Standard +Sean +\begin_inset Formula $m,n\in\mathbb{N}^{*}$ +\end_inset + + y +\begin_inset Formula ${\cal C}_{m}$ +\end_inset + + y +\begin_inset Formula ${\cal C}_{n}$ +\end_inset + + las bases canónicas respectivas de los +\begin_inset Formula $A$ +\end_inset + +-módulos libres +\begin_inset Formula $A^{m}$ +\end_inset + + y +\begin_inset Formula $A^{n}$ +\end_inset + +, +\begin_inset Formula $(f\mapsto M_{{\cal C}_{m}{\cal C}_{n}}(f)):\text{Hom}_{A}(A^{n},A^{m})\to{\cal M}_{m\times n}(A)$ +\end_inset + + es un isomorfismo de +\begin_inset Formula $A$ +\end_inset + +-módulos con inversa +\begin_inset Formula $C\mapsto v\mapsto Cv$ +\end_inset + +. + +\series bold +Demostración: +\series default + Sean +\begin_inset Formula ${\cal C}_{n}\eqqcolon(e_{1},\dots,e_{n})$ +\end_inset + + y +\begin_inset Formula ${\cal C}_{m}\eqqcolon(f_{1},\dots,f_{m})$ +\end_inset + +, toda +\begin_inset Formula $f\in\text{Hom}_{A}(A^{n},A^{m})$ +\end_inset + + viene dada por los valores que le asigna a los +\begin_inset Formula $e_{i}$ +\end_inset + +, que se pueden expresar respecto a los +\begin_inset Formula $f_{j}$ +\end_inset + + dando lugar a +\begin_inset Formula $M\coloneqq M_{{\cal C}_{m}{\cal C}_{n}}(f)$ +\end_inset + + cuyas columnas son los +\begin_inset Formula $f(e_{i})$ +\end_inset + +, pero claramente +\begin_inset Formula $Me_{i}$ +\end_inset + + es la +\begin_inset Formula $i$ +\end_inset + +-ésima columna de +\begin_inset Formula $M$ +\end_inset + +, y recíprocamente, si +\begin_inset Formula $M\in{\cal M}_{m\times n}(A)$ +\end_inset + + y +\begin_inset Formula $f$ +\end_inset + + viene dada por +\begin_inset Formula $f(v)\coloneqq Mv$ +\end_inset + +, las columnas de +\begin_inset Formula $M_{{\cal C}_{m}{\cal C}_{n}}(f)$ +\end_inset + + son los +\begin_inset Formula $Me_{i}$ +\end_inset + + que son las columnas de +\begin_inset Formula $M$ +\end_inset + +. + Que es un isomorfismo es claro tomando +\begin_inset Formula $(b_{ij}\coloneqq\sum_{k}a_{k}e_{k}\mapsto a_{i}f_{j})_{i,j}$ +\end_inset + + como base de +\begin_inset Formula $\text{Hom}_{A}(A^{n},A^{m})$ +\end_inset + + y viendo que conserva combinaciones lineales de los +\begin_inset Formula $b_{ij}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset Formula $\text{GL}_{s}(K)\coloneqq\{A\in{\cal M}_{s}(K)\mid\det A\neq0\}$ +\end_inset + +. + Dada +\begin_inset Formula $C\in{\cal M}_{m\times n}(A)$ +\end_inset + +, llamamos +\series bold + +\begin_inset Formula $A$ +\end_inset + +-módulo asociado a +\begin_inset Formula $C$ +\end_inset + + +\series default +, +\begin_inset Formula $M(C)$ +\end_inset + +, a +\begin_inset Formula $\frac{A^{m}}{\{Cv\}_{v\in A^{n}}}$ +\end_inset + +. + +\begin_inset Formula $B,C\in{\cal M}_{m\times n}(A)$ +\end_inset + + son +\series bold +equivalentes +\series default + si existen +\begin_inset Formula $P\in\text{GL}_{m}(A)$ +\end_inset + + y +\begin_inset Formula $Q\in\text{GL}_{n}(A)$ +\end_inset + + con +\begin_inset Formula $C=PBQ$ +\end_inset + +, en cuyo caso +\begin_inset Formula $M(B)\cong M(C)$ +\end_inset + +. + +\series bold +Demostración: +\series default + Se tiene +\begin_inset Formula $PB=CQ^{-1}$ +\end_inset + +, luego llamando +\begin_inset Formula $f_{C}:A^{n}\to A^{m}$ +\end_inset + + al homomorfismo +\begin_inset Formula $f_{C}(v)\coloneqq Cv$ +\end_inset + +, +\begin_inset Formula $f_{P}\circ f_{B}=f_{C}\circ f_{Q^{-1}}$ +\end_inset + +. + Definiendo el homomorfismo +\begin_inset Formula $\psi:M(B)\to M(C)$ +\end_inset + + como +\begin_inset Formula $\psi(\overline{a})=\overline{f_{P}(a)}$ +\end_inset + +, +\begin_inset Formula $\psi$ +\end_inset + + está bien definido porque +\begin_inset Formula $a\in\text{Im}f_{B}\implies f_{P}(a)\in\text{Im}(f_{P}\circ f_{B})=\text{Im}(f_{C}\circ f_{Q^{-1}})=\text{Im}f_{C}$ +\end_inset + +, pero el homomorfismo +\begin_inset Formula $\phi:M(C)\to M(B)$ +\end_inset + + dado por +\begin_inset Formula $\phi(\overline{c})\coloneqq\overline{f_{P^{-1}}(c)}$ +\end_inset + + también está bien definido porque +\begin_inset Formula $c\in\text{Im}f_{C}\implies f_{P^{-1}}(c)\in\text{Im}(f_{P^{-1}}\circ f_{C})=\text{Im}(f_{P^{-1}}\circ f_{C}\circ f_{Q^{-1}})=\text{Im}(f_{P})$ +\end_inset + +, y +\begin_inset Formula $\phi=\psi^{-1}$ +\end_inset + +. +\end_layout + +\begin_layout Standard +Una +\series bold +operación +\series default + o +\series bold +transformación elemental por filas +\series default + o +\series bold +columnas +\series default + en +\begin_inset Formula $C\in{\cal M}_{m\times n}(A)$ +\end_inset + + consiste en intercambiar dos filas o columnas de +\begin_inset Formula $C$ +\end_inset + +, multiplicar una por un +\begin_inset Formula $\alpha\in A^{*}$ +\end_inset + + o sumarle a una otra multiplicada por un +\begin_inset Formula $\alpha\in A$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +begin{reminder}{AlgL} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Llamamos +\series bold +matriz elemental +\series default + de tamaño +\begin_inset Formula $n$ +\end_inset + + a toda matriz obtenida al efectuar una operación elemental [...] en +\begin_inset Formula $I_{n}$ +\end_inset + +. + [...] Si +\begin_inset Formula $B$ +\end_inset + + se obtiene al realizar una operación elemental por filas en +\begin_inset Formula $A$ +\end_inset + + y +\begin_inset Formula $E$ +\end_inset + + al realizar la misma en +\begin_inset Formula $I_{m}$ +\end_inset + +, entonces +\begin_inset Formula $B=EA$ +\end_inset + +. + [...] Si +\begin_inset Formula $B$ +\end_inset + + se obtiene de aplicar una operación elemental por columnas en +\begin_inset Formula $A$ +\end_inset + + y +\begin_inset Formula $E$ +\end_inset + + al aplicarla a +\begin_inset Formula $I_{n}$ +\end_inset + +, entonces +\begin_inset Formula $B=AE$ +\end_inset + +. + Así, realizar una serie de estas operaciones en una matriz equivale a multiplic +arla por uno o ambos lados por un producto de matrices elementales, el cual + es invertible. +\end_layout + +\begin_layout Standard +\begin_inset ERT +status open + +\begin_layout Plain Layout + + +\backslash +end{reminder} +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Standard +Las matrices elementales son las mismas por filas que por columnas. + Si +\begin_inset Formula $B,C\in{\cal M}_{m\times n}(A)$ +\end_inset + + y +\begin_inset Formula $C$ +\end_inset + + se puede obtener aplicando a +\begin_inset Formula $B$ +\end_inset + + una cantidad finita de transformaciones elementales por filas y por columnas, + entonces +\begin_inset Formula $B$ +\end_inset + + y +\begin_inset Formula $C$ +\end_inset + + son equivalentes, pues aplicar transformaciones por filas y columnas a + +\begin_inset Formula $B$ +\end_inset + + equivale a multiplicarla a izquierda y derecha por matrices invertibles. +\end_layout + \end_body \end_document |