Errata ANM

1 files changed, 98 insertions, 212 deletions

diff --git a/anm/n1.lyx b/anm/n1.lyx
index 96438de..6dd1313 100644
--- a/anm/n1.lyx
+++ b/anm/n1.lyx
@@ -278,7 +278,7 @@ traza
  a
 \begin_inset Formula 
 \[
-\text{tr}A:=\sum_{k=1}^{n}a_{kk}.
+\text{tr}A:=\sum_{k=1}^{n}A_{kk}.
 \]
 
 \end_inset
@@ -395,7 +395,7 @@ sistema
 \begin_inset Formula $n$
 \end_inset
 
- incógnitas es un sistema de la forma
+ incógnitas es uno de la forma
 \begin_inset Formula 
 \[
 \left\{ \begin{aligned}a_{11}x_{1}+\dots+a_{1n}x_{n} & =b_{1},\\
@@ -522,15 +522,15 @@ Una
 \series bold
 base
 \series default
- de un espacio vectorial 
-\begin_inset Formula $E$
+ de un 
+\begin_inset Formula $\mathbb{K}$
 \end_inset
 
- de dimensión finita sobre un cuerpo 
-\begin_inset Formula $\mathbb{K}$
+-espacio vectorial 
+\begin_inset Formula $E$
 \end_inset
 
- es una tupla 
+ de dimensión finita es una tupla 
 \begin_inset Formula $(v_{1},\dots,v_{n})$
 \end_inset
 
@@ -563,7 +563,7 @@ con
 \begin_inset Formula $(x_{1},\dots,x_{n})\in\mathbb{K}^{n}$
 \end_inset
 
-, y por tanto con la correspondiente matriz columna.
+, y con la correspondiente matriz columna.
 \end_layout
 
 \begin_layout Standard
@@ -584,7 +584,7 @@ producto escalar
 \end_inset
 
  bilineal simétrica tal que 
-\begin_inset Formula $\forall f\in E\setminus\{0\},\langle f,f\rangle>0$
+\begin_inset Formula $\forall f\in E\setminus0,\langle f,f\rangle>0$
 \end_inset
 
 .
@@ -686,169 +686,6 @@ ortogonales
 \end_inset
 
 .
- 
-\end_layout
-
-\begin_layout Standard
-Una 
-\series bold
-norma
-\series default
- en un 
-\begin_inset Formula $\mathbb{R}$
-\end_inset
-
--espacio vectorial 
-\begin_inset Formula $E$
-\end_inset
-
- es una aplicación 
-\begin_inset Formula $\Vert\cdot\Vert:E\to\mathbb{K}$
-\end_inset
-
- tal que 
-\begin_inset Formula $\forall v,w\in E,t\in\mathbb{R}:$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\Vert tv\Vert=|t|\Vert v\Vert$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $\Vert v+w\Vert\leq\Vert v\Vert+\Vert w\Vert$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
-\begin_inset Formula $v\neq0\implies\Vert v\Vert>0$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Llamamos 
-\series bold
-norma euclídea
-\series default
- en 
-\begin_inset Formula $\mathbb{R}^{n}$
-\end_inset
-
- a 
-\begin_inset Formula $\Vert v\Vert:=\sqrt{\langle v,v\rangle}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Sean 
-\begin_inset Formula $f:V\to W$
-\end_inset
-
- una aplicación lineal y 
-\begin_inset Formula ${\cal B}:=(v_{1},\dots,v_{n})$
-\end_inset
-
- y 
-\begin_inset Formula ${\cal B}':=(w_{1},\dots,w_{m})$
-\end_inset
-
- bases respectivas de 
-\begin_inset Formula $V$
-\end_inset
-
- y 
-\begin_inset Formula $W$
-\end_inset
-
-, si
-\begin_inset Formula 
-\[
-\left\{ \begin{aligned}f(v_{1}) & =a_{11}w_{1}+\dots+a_{m1}w_{1},\\
- & \vdots\\
-f(v_{m}) & =a_{1n}w_{1}+\dots+a_{mn}w_{m},
-\end{aligned}
-\right.
-\]
-
-\end_inset
-
-llamamos 
-\series bold
-matriz asociada
-\series default
- a 
-\begin_inset Formula $f$
-\end_inset
-
- con respecto de las bases 
-\begin_inset Formula ${\cal B}$
-\end_inset
-
- y 
-\begin_inset Formula ${\cal B}'$
-\end_inset
-
- a 
-\begin_inset Formula $(a_{ij})_{1\leq i\leq m}^{1\leq j\leq n}$
-\end_inset
-
-.
- Dadas dos aplicaciones lineales 
-\begin_inset Formula $U\overset{f}{\to}V\overset{g}{\to}W$
-\end_inset
-
-, 
-\begin_inset Formula $g\circ f$
-\end_inset
-
- también es lineal, y si 
-\begin_inset Formula $U$
-\end_inset
-
-, 
-\begin_inset Formula $V$
-\end_inset
-
- y 
-\begin_inset Formula $W$
-\end_inset
-
- son de dimensión finita y 
-\begin_inset Formula $f$
-\end_inset
-
- y 
-\begin_inset Formula $g$
-\end_inset
-
- tienen matrices respectivas 
-\begin_inset Formula $A$
-\end_inset
-
- y 
-\begin_inset Formula $B$
-\end_inset
-
-, 
-\begin_inset Formula $g\circ f$
-\end_inset
-
- tiene matriz 
-\begin_inset Formula $BA$
-\end_inset
-
- respecto de las mismas bases.
 \end_layout
 
 \begin_layout Section
@@ -1443,11 +1280,11 @@ El
 cociente de Rayleigh
 \series default
  de una matriz 
-\begin_inset Formula $A\in{\cal M}_{n}$
+\begin_inset Formula $A\in{\cal M}_{n}(\mathbb{C})$
 \end_inset
 
  es una aplicación 
-\begin_inset Formula $R_{A}:\mathbb{C}^{n}\setminus\{0\}\to\mathbb{C}$
+\begin_inset Formula $R_{A}:\mathbb{C}^{n}\setminus0\to\mathbb{C}$
 \end_inset
 
  dada por
@@ -1474,7 +1311,7 @@ Sean
 \begin_inset Formula $A\in{\cal M}_{n}$
 \end_inset
 
- es hermitiana con valores propios 
+ hermitiana con valores propios 
 \begin_inset Formula $\lambda_{1}\leq\dots\leq\lambda_{n}$
 \end_inset
 
@@ -1498,7 +1335,11 @@ Sean
 \begin_inset Formula $k$
 \end_inset
 
-, 
+ (
+\begin_inset Formula $E_{0}=\{0\}$
+\end_inset
+
+) y 
 \begin_inset Formula ${\cal S}_{k}$
 \end_inset
 
@@ -1510,14 +1351,6 @@ Sean
 \begin_inset Formula $k$
 \end_inset
 
-, 
-\begin_inset Formula $E_{0}:=\{0\}$
-\end_inset
-
- y 
-\begin_inset Formula ${\cal S}_{k}:=\{E_{0}\}$
-\end_inset
-
 .
  Entonces, para 
 \begin_inset Formula $1\leq k\leq n$
@@ -1750,11 +1583,15 @@ Sea
 \begin_inset Formula $E$
 \end_inset
 
- un 
-\begin_inset Formula $\mathbb{K}$
+ un espacio vectorial sobre 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{C}$
 \end_inset
 
--espacio vectorial, una 
+, una 
 \series bold
 norma
 \series default
@@ -1892,7 +1729,19 @@ norma matricial
 \begin_inset Formula ${\cal M}_{n}(\mathbb{K})$
 \end_inset
 
- es una que cumple 
+, donde 
+\begin_inset Formula $\mathbb{K}$
+\end_inset
+
+ es 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ o 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+, es una que cumple 
 \begin_inset Formula $\forall A,B\in{\cal M}_{n}(\mathbb{K}),\Vert AB\Vert\leq\Vert A\Vert\Vert B\Vert$
 \end_inset
 
@@ -1907,7 +1756,63 @@ norma matricial
 
 , llamamos 
 \series bold
-norma matricial subordinada
+norma matricial sub
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+or
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+di
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+na
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+da
 \series default
  a la norma 
 \begin_inset Formula $\Vert\cdot\Vert$
@@ -1920,7 +1825,7 @@ norma matricial subordinada
  dada por
 \begin_inset Formula 
 \[
-\Vert A\Vert:=\sup\left\{ \frac{\Vert Ax\Vert}{\Vert x\Vert}\right\} _{x\in\mathbb{K}^{n}\setminus\{0\}}=\sup\left\{ \frac{\Vert Ax\Vert}{\Vert x\Vert}\right\} _{\Vert x\Vert\leq1}=\sup\left\{ \Vert Ax\Vert\right\} _{\Vert x\Vert=1}.
+\Vert A\Vert:=\sup_{x\in\mathbb{K}^{n}\setminus\{0\}}\frac{\Vert Ax\Vert}{\Vert x\Vert}=\sup_{\Vert x\Vert\leq1}\frac{\Vert Ax\Vert}{\Vert x\Vert}=\sup_{\Vert x\Vert=1}\Vert Ax\Vert.
 \]
 
 \end_inset
@@ -1941,13 +1846,6 @@ Entonces, para
 \end_layout
 
 \begin_layout Standard
-\begin_inset Newpage pagebreak
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
 Sea 
 \begin_inset Formula $A:=(a_{ij})_{ij}\in{\cal M}_{n}(\mathbb{C})$
 \end_inset
@@ -2139,7 +2037,7 @@ norma euclídea
 \begin_inset Formula $\Vert\cdot\Vert_{2}$
 \end_inset
 
- y 
+, y 
 \begin_inset Formula $\Vert A\Vert_{2}\leq\Vert A\Vert_{E}\leq\sqrt{n}\Vert A\Vert_{2}$
 \end_inset
 
@@ -2497,7 +2395,7 @@ Sean
 \end_inset
 
  invertible, 
-\begin_inset Formula $0\neq b\in\mathbb{K}^{n}$
+\begin_inset Formula $b\in\mathbb{K}^{n}\setminus0$
 \end_inset
 
  y 
@@ -2682,7 +2580,7 @@ Llamamos
 \end_layout
 
 \begin_layout Enumerate
-\begin_inset Formula $\forall\alpha\in\mathbb{K}\setminus\{0\},\text{cond}(\alpha A)=\text{cond}A$
+\begin_inset Formula $\forall\alpha\in\mathbb{K}\setminus0,\text{cond}(\alpha A)=\text{cond}A$
 \end_inset
 
 .
@@ -2733,18 +2631,6 @@ Si
 \end_layout
 
 \begin_layout Enumerate
-Si 
-\begin_inset Formula $A$
-\end_inset
-
- es unitaria, 
-\begin_inset Formula $\text{cond}_{2}U=1$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Enumerate
 Sea 
 \begin_inset Formula $U$
 \end_inset