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-rw-r--r--aalg/n4.lyx104
1 files changed, 52 insertions, 52 deletions
diff --git a/aalg/n4.lyx b/aalg/n4.lyx
index 96b456a..c6a789e 100644
--- a/aalg/n4.lyx
+++ b/aalg/n4.lyx
@@ -202,7 +202,7 @@ Sean
\end_inset
y
-\begin_inset Formula $A:=(a_{ij}:=\langle e_{i},e_{j}\rangle)\in{\cal M}_{n}(\mathbb{K})$
+\begin_inset Formula $A\coloneqq (a_{ij}\coloneqq \langle e_{i},e_{j}\rangle)\in{\cal M}_{n}(\mathbb{K})$
\end_inset
, entonces si
@@ -277,7 +277,7 @@ forma cuadrática
\end_inset
dada por
-\begin_inset Formula $\langle u,v\rangle:=\frac{1}{2}(q(u+v)-q(u)-q(v))$
+\begin_inset Formula $\langle u,v\rangle\coloneqq \frac{1}{2}(q(u+v)-q(u)-q(v))$
\end_inset
es una forma bilineal simétrica en
@@ -315,7 +315,7 @@ Llamamos
que asocia a cada forma cuadrática su forma polar es biyectiva y su inversa
asocia a cada forma bilineal simétrica la forma cuadrática dada por
-\begin_inset Formula $q(u):=\langle u,u\rangle$
+\begin_inset Formula $q(u)\coloneqq \langle u,u\rangle$
\end_inset
.
@@ -328,7 +328,7 @@ Demostración:
\end_inset
y
-\begin_inset Formula $q(u):=\langle u,u\rangle$
+\begin_inset Formula $q(u)\coloneqq \langle u,u\rangle$
\end_inset
, es claro que
@@ -369,7 +369,7 @@ Sean ahora
\begin_layout Standard
Esta correspondencia permite asociar una matriz
-\begin_inset Formula $A:=(a_{ij})\in{\cal M}_{n}(\mathbb{K})$
+\begin_inset Formula $A\coloneqq (a_{ij})\in{\cal M}_{n}(\mathbb{K})$
\end_inset
a una forma cuadrática
@@ -409,11 +409,11 @@ Sean
\end_inset
un espacio bilineal,
-\begin_inset Formula ${\cal C}:=(u_{1},\dots,u_{n})$
+\begin_inset Formula ${\cal C}\coloneqq (u_{1},\dots,u_{n})$
\end_inset
y
-\begin_inset Formula ${\cal B}:=(v_{1},\dots,v_{n})$
+\begin_inset Formula ${\cal B}\coloneqq (v_{1},\dots,v_{n})$
\end_inset
bases de
@@ -425,11 +425,11 @@ Sean
\end_inset
tiene matrices respectivas
-\begin_inset Formula $A:=(a_{ij})$
+\begin_inset Formula $A\coloneqq (a_{ij})$
\end_inset
y
-\begin_inset Formula $B:=(b_{ij})$
+\begin_inset Formula $B\coloneqq (b_{ij})$
\end_inset
,
@@ -751,7 +751,7 @@ A\sim A'\iff\langle\cdot\rangle\sim\langle\cdot\rangle'\iff(V,\langle\cdot\rangl
\end_inset
tienen la misma matriz asociada
-\begin_inset Formula $C:=(c_{ij})$
+\begin_inset Formula $C\coloneqq (c_{ij})$
\end_inset
, entonces
@@ -775,11 +775,11 @@ A\sim A'\iff\langle\cdot\rangle\sim\langle\cdot\rangle'\iff(V,\langle\cdot\rangl
\end_inset
una isometría y
-\begin_inset Formula ${\cal B}:=(v_{1},\dots,v_{n})$
+\begin_inset Formula ${\cal B}\coloneqq (v_{1},\dots,v_{n})$
\end_inset
, entonces
-\begin_inset Formula ${\cal B}':=(f(v_{1}),\dots,f(v_{n}))$
+\begin_inset Formula ${\cal B}'\coloneqq (f(v_{1}),\dots,f(v_{n}))$
\end_inset
es una base de
@@ -791,7 +791,7 @@ A\sim A'\iff\langle\cdot\rangle\sim\langle\cdot\rangle'\iff(V,\langle\cdot\rangl
\end_inset
, ambas formas bilineales tienen la misma matriz
-\begin_inset Formula $C:=(c_{ij})$
+\begin_inset Formula $C\coloneqq (c_{ij})$
\end_inset
, y entonces
@@ -827,7 +827,7 @@ subespacio ortogonal
\end_inset
al subespacio
-\begin_inset Formula $E^{\bot}:=\{v\in V\mid \forall e\in E,\langle v,e\rangle=0\}$
+\begin_inset Formula $E^{\bot}\coloneqq \{v\in V\mid \forall e\in E,\langle v,e\rangle=0\}$
\end_inset
.
@@ -865,7 +865,7 @@ radical
\end_inset
a
-\begin_inset Formula $Rad(V):=V^{\bot}$
+\begin_inset Formula $Rad(V)\coloneqq V^{\bot}$
\end_inset
.
@@ -1022,7 +1022,7 @@ Demostración:
.
Sea
-\begin_inset Formula ${\cal B}:=(e_{1},\dots,e_{m})$
+\begin_inset Formula ${\cal B}\coloneqq (e_{1},\dots,e_{m})$
\end_inset
una base de
@@ -1062,12 +1062,12 @@ x_{m}
\end_inset
tiene solución única y
-\begin_inset Formula $x:=\sum x_{i}e_{i}\in E$
+\begin_inset Formula $x\coloneqq \sum x_{i}e_{i}\in E$
\end_inset
.
Sea
-\begin_inset Formula $v:=u-x$
+\begin_inset Formula $v\coloneqq u-x$
\end_inset
,
@@ -1147,7 +1147,7 @@ Demostración:
\end_inset
no isótropo y, si
-\begin_inset Formula $E:=<e_{1}>$
+\begin_inset Formula $E\coloneqq <e_{1}>$
\end_inset
,
@@ -1237,7 +1237,7 @@ Si
\end_inset
, basta tomar
-\begin_inset Formula $P:=E_{1}^{t}\cdots E_{k}^{t}$
+\begin_inset Formula $P\coloneqq E_{1}^{t}\cdots E_{k}^{t}$
\end_inset
.
@@ -1954,11 +1954,11 @@ Reescribir
\end_inset
Hacer el cambio
-\begin_inset Formula $x'_{1}:=x_{1}+\frac{p(x_{2},\dots,x_{n})}{2a_{11}}$
+\begin_inset Formula $x'_{1}\coloneqq x_{1}+\frac{p(x_{2},\dots,x_{n})}{2a_{11}}$
\end_inset
y
-\begin_inset Formula $x'_{j}:=x_{j},j\neq1$
+\begin_inset Formula $x'_{j}\coloneqq x_{j},j\neq1$
\end_inset
@@ -2040,7 +2040,7 @@ Demostración:
\end_inset
y
-\begin_inset Formula $U:=V_{(\alpha_{1})}\oplus\dots\oplus V_{(\alpha_{m})}$
+\begin_inset Formula $U\coloneqq V_{(\alpha_{1})}\oplus\dots\oplus V_{(\alpha_{m})}$
\end_inset
, siendo
@@ -2165,7 +2165,7 @@ rango
\end_inset
a
-\begin_inset Formula $\text{rg}(\langle\cdot\rangle):=\text{rg}(A)=\dim(V)-\dim Rad(\langle\cdot\rangle)$
+\begin_inset Formula $\text{rg}(\langle\cdot\rangle)\coloneqq \text{rg}(A)=\dim(V)-\dim Rad(\langle\cdot\rangle)$
\end_inset
.
@@ -2239,7 +2239,7 @@ Demostración:
\end_inset
con
-\begin_inset Formula $\lambda:=|P|$
+\begin_inset Formula $\lambda\coloneqq |P|$
\end_inset
.
@@ -2309,11 +2309,11 @@ Demostración:
\end_inset
es
-\begin_inset Formula $D:=\text{diag}(d_{1},\dots,d_{m},0,\dots,0)$
+\begin_inset Formula $D\coloneqq \text{diag}(d_{1},\dots,d_{m},0,\dots,0)$
\end_inset
, siendo
-\begin_inset Formula $m:=\text{rg}(\langle\cdot\rangle)$
+\begin_inset Formula $m\coloneqq \text{rg}(\langle\cdot\rangle)$
\end_inset
, con
@@ -2434,7 +2434,7 @@ positivos
.
A los elementos de
-\begin_inset Formula $-P:=\{-x\}_{x\in P}$
+\begin_inset Formula $-P\coloneqq \{-x\}_{x\in P}$
\end_inset
los llamamos
@@ -2568,7 +2568,7 @@ Las mismas definiciones se aplican a una forma cuadrática.
\end_inset
,
-\begin_inset Formula $A:=(a_{ij})$
+\begin_inset Formula $A\coloneqq (a_{ij})$
\end_inset
la matriz de
@@ -2576,7 +2576,7 @@ Las mismas definiciones se aplican a una forma cuadrática.
\end_inset
en cierta base
-\begin_inset Formula ${\cal C}:=(e_{1},\dots,e_{n})$
+\begin_inset Formula ${\cal C}\coloneqq (e_{1},\dots,e_{n})$
\end_inset
y definimos
@@ -2608,7 +2608,7 @@ a_{21} & a_{22}
Demostración:
\series default
Sea
-\begin_inset Formula $E:=<e_{1},\dots,e_{n-1}>$
+\begin_inset Formula $E\coloneqq <e_{1},\dots,e_{n-1}>$
\end_inset
, la matriz de
@@ -2675,7 +2675,7 @@ Tenemos
.
Sea
-\begin_inset Formula $\lambda:=|P|$
+\begin_inset Formula $\lambda\coloneqq |P|$
\end_inset
,
@@ -2691,7 +2691,7 @@ Tenemos
\end_inset
en la base
-\begin_inset Formula $(e_{1},\dots,e_{n-1},w:=\lambda v)$
+\begin_inset Formula $(e_{1},\dots,e_{n-1},w\coloneqq \lambda v)$
\end_inset
es como
@@ -2773,11 +2773,11 @@ teorema de Sylvester
es definida positiva, definida negativa y nula, respectivamente.
Además,
-\begin_inset Formula $p:=\dim(V_{+})$
+\begin_inset Formula $p\coloneqq \dim(V_{+})$
\end_inset
y
-\begin_inset Formula $m:=\dim(V_{-})$
+\begin_inset Formula $m\coloneqq \dim(V_{-})$
\end_inset
son únicos, y al par
@@ -2798,7 +2798,7 @@ signatura
Demostración:
\series default
Sea
-\begin_inset Formula ${\cal C}:=(e_{1},\dots,e_{n})$
+\begin_inset Formula ${\cal C}\coloneqq (e_{1},\dots,e_{n})$
\end_inset
una base de
@@ -3095,7 +3095,7 @@ Si
\end_inset
, y entonces definimos
-\begin_inset Formula $t(w):=-w$
+\begin_inset Formula $t(w)\coloneqq -w$
\end_inset
y vemos que
@@ -3111,11 +3111,11 @@ Como
teorema
\series default
, si
-\begin_inset Formula $D_{1}:=\text{diag}(a_{1},\dots,a_{r},b_{r+1},\dots,b_{n})$
+\begin_inset Formula $D_{1}\coloneqq \text{diag}(a_{1},\dots,a_{r},b_{r+1},\dots,b_{n})$
\end_inset
y
-\begin_inset Formula $D_{2}:=\text{diag}(a_{1},\dots,a_{r},c_{r+1},\dots,c_{n})$
+\begin_inset Formula $D_{2}\coloneqq \text{diag}(a_{1},\dots,a_{r},c_{r+1},\dots,c_{n})$
\end_inset
son matrices con
@@ -3190,7 +3190,7 @@ Demostración:
\end_inset
y
-\begin_inset Formula $E:=<s(u_{2}),\dots,s(u_{n})>=<s(u_{1})>^{\bot}=<v_{1}>^{\bot}=<v_{2},\dots,v_{n}>$
+\begin_inset Formula $E\coloneqq <s(u_{2}),\dots,s(u_{n})>=<s(u_{1})>^{\bot}=<v_{1}>^{\bot}=<v_{2},\dots,v_{n}>$
\end_inset
.
@@ -3309,11 +3309,11 @@ Demostración:
\end_inset
, si
-\begin_inset Formula $D_{1}:=\text{diag}(a_{1},\dots,a_{r},b_{r+1},\dots,b_{n})$
+\begin_inset Formula $D_{1}\coloneqq \text{diag}(a_{1},\dots,a_{r},b_{r+1},\dots,b_{n})$
\end_inset
y
-\begin_inset Formula $D_{2}:=\text{diag}(a_{1},\dots,a_{r},c_{r+1},\dots,c_{n})$
+\begin_inset Formula $D_{2}\coloneqq \text{diag}(a_{1},\dots,a_{r},c_{r+1},\dots,c_{n})$
\end_inset
son las matrices de
@@ -3430,7 +3430,7 @@ Sean
\end_inset
es una base y
-\begin_inset Formula $v':=\frac{v}{\langle u,v\rangle}$
+\begin_inset Formula $v'\coloneqq \frac{v}{\langle u,v\rangle}$
\end_inset
, la matriz de
@@ -3453,12 +3453,12 @@ A:=\left(\begin{array}{cc}
\end_inset
con
-\begin_inset Formula $a:=\langle v',v'\rangle$
+\begin_inset Formula $a\coloneqq \langle v',v'\rangle$
\end_inset
.
Sea ahora
-\begin_inset Formula $w:=xu+v'$
+\begin_inset Formula $w\coloneqq xu+v'$
\end_inset
tal que
@@ -3498,7 +3498,7 @@ B:=\left(\begin{array}{cc}
\end_inset
con
-\begin_inset Formula $b:=\langle w',w'\rangle$
+\begin_inset Formula $b\coloneqq \langle w',w'\rangle$
\end_inset
.
@@ -3576,7 +3576,7 @@ Si identificamos los vectores con sus coordenadas respecto a la base en
\end_inset
es isótropo no nulo y, si hubiera un
-\begin_inset Formula $v:=(v_{1},v_{2})$
+\begin_inset Formula $v\coloneqq (v_{1},v_{2})$
\end_inset
con
@@ -3744,7 +3744,7 @@ Demostración:
es anisótropo.
Si
-\begin_inset Formula $n:=\dim(V)\geq2$
+\begin_inset Formula $n\coloneqq \dim(V)\geq2$
\end_inset
y
@@ -3827,12 +3827,12 @@ cónica proyectiva
\end_inset
, o de formas cuadráticas no nulas de dimensión 3, bajo la relación
-\begin_inset Formula $q\sim q':\iff\exists\lambda\in\mathbb{K}\backslash\{0\}\mid q'=\lambda q$
+\begin_inset Formula $q\sim q':\iff\exists\lambda\in\mathbb{K}\backslash\{0\}:q'=\lambda q$
\end_inset
.
Escribimos
-\begin_inset Formula ${\cal C}_{q}:=[q]$
+\begin_inset Formula ${\cal C}_{q}\coloneqq [q]$
\end_inset
, y la identificamos con el conjunto de puntos
@@ -3975,7 +3975,7 @@ recta polar
\end_inset
a
-\begin_inset Formula $r_{P}:=\{X\in\mathbb{P}^{2}(\mathbb{K})\mid [P]^{t}\overline{A}[X]=0\}$
+\begin_inset Formula $r_{P}\coloneqq \{X\in\mathbb{P}^{2}(\mathbb{K})\mid [P]^{t}\overline{A}[X]=0\}$
\end_inset
, y decimos que
@@ -4035,7 +4035,7 @@ Una cónica es
no degenerada
\series default
si
-\begin_inset Formula $\Delta:=|\overline{A}|\neq0$
+\begin_inset Formula $\Delta\coloneqq |\overline{A}|\neq0$
\end_inset
.
generated by cgit v1.2.3 (git 2.46.0) at 2025-10-31 09:19:21 +0000