problem with old hebrew latex files in iso 8859-8 coding

problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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Hello 

A friend of mine posses a lot of hebrew latex files which still use
8859-8 coding. Understandable he wants to switch to UTF8 and xelatex.

The problem is that while his files are displayed perfectly in texstudio
(having iso 8859-8 enabled), which means the system has the requires
fonts installed, emacs fails to do so. It simply does not display the
hebrew chars with the correct fonts.

What also works is to run 

iconv -f ISO8859-8 -t UTF-8 file1.txt file2.txt

The resulting file file2.txt is displayed perfectly in GNU emacs.



What does work is to open the file in GNU emacs, save it as UTF-8, the
resulting file is still not correctly displayed, I even tried to save
the file as 8859-8 but then I received the following error



,----
| These default coding systems were tried to encode text
| in the buffer ‘doc2.txt’:
|   (hebrew-iso-8bit-unix (1 . 233) (2 . 228) (3 . 233) (7 . 238) (8 .
|   248) (9 . 231) (10 . 225) (12 . 229) (14 . 232) ...)
| However, each of them encountered characters it couldn’t encode:
|   hebrew-iso-8bit-unix cannot encode these: é ä é î ø ç á å è å ...
| 
| Click on a character (or switch to this window by ‘C-x o’
| and select the characters by RET) to jump to the place it appears,
| where ‘C-u C-x =’ will give information about it.
| 
| Select one of the safe coding systems listed below,
| or cancel the writing with C-g and edit the buffer
|    to remove or modify the problematic characters,
| or specify any other coding system (and risk losing
|    the problematic characters).
| 
|   utf-8 iso-8859-1 euc-jis-2004 euc-jp iso-2022-jp-2004 next cp858
|   cp850 windows-1252 iso-8859-15 gb18030 utf-7 utf-16
|   utf-16be-with-signature utf-16le-with-signature utf-16be utf-16le
|   iso-2022-7bit utf-8-auto utf-8-with-signature eucjp-ms utf-8-hfs
|   japanese-shift-jis-2004 japanese-iso-7bit-1978-irv ibm1047
|   utf-7-imap utf-8-emacs prefer-utf-8
`----

I attach the file, once as a original file, once with the latex commands
removed.

Any help is strongly appreciated.

Regards

Uwe Brauer 


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éäé 

îøçá å÷èåøé ëàùø 

å 

áñéñ ì  
äåëéçå àå äôøéëå àú äèòðåú äáàåú. 


ìëì å÷èåø 

ä÷áåöä 

áñéñ ì 

[-- Attachment #3: junk.tex --]
[-- Type: text/x-tex, Size: 31649 bytes --]

\documentclass[11pt]{article}
%\usepackage[8859-8]{inputenc}
\usepackage[english,hebrew]{babel}
\usepackage{color}
%\usepackage{hebcal}
\usepackage{culmus}
\usepackage{courier}
\usepackage{ulem}
%\usepackage[HE8,OT1]{fontenc}
%\usepackage{ccfonts}
%\usepackage{float}
% \usepackage{color}
\usepackage{theorem}
\usepackage{fancyhdr}
\usepackage{esvect}
\usepackage{graphicx}
\usepackage{epstopdf}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usetikzlibrary{calc}
%\usepackage{url}
%\usepackage{listings}
%\usepackage{a4wide}
%\renewcommand{\baselinestretch}{1.5}
%\usepackage{graphicx,epsf}

%\renewcommand{\labelenumi}{\alph{enumi}.}
\usepackage{amssymb,amsfonts,amsmath}
\usepackage{pgffor, ifthen}
\newcommand{\notes}[3][\empty]{%
	\noindent \vspace{10pt}\\
	\foreach \n in {1,...,#2}{%
		\ifthenelse{\equal{#1}{\empty}}
		{\rule{#3}{0.5pt}\\}
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%%    \noindent\refstepcounter{parnum}%
%%     \makebox[\parindent][l]{\textbf{\arabic{parnum}.}}}
% Use a generous paragraph indent so numbers can be fit inside the
% indentation space.
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\setlength{\parindent}{2em}

\setlength{\textheight}{24.5cm}
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\newcommand{\setB}{{\mathord{\mathbb B}}}
\newcommand{\setC}{{\mathord{\mathbb C}}}
\newcommand{\setF}{{\mathord{\mathbb F}}}
\newcommand{\setN}{{\mathord{\mathbb N}}}
\newcommand{\setQ}{{\mathord{\mathbb Q}}}
\newcommand{\setR}{{\mathord{\mathbb R}}}
\newcommand{\setZ}{{\mathord{\mathbb Z}}}
\newcommand{\nk}[1] {\textcolor{red}{)#1 ð÷'(}}
\newcommand{\red}[1] {\textcolor{red}{#1}}
%
\newcommand{\br}[1]{\left[{#1}\right]} %square bracets
\newcommand{\brf}[1]{\left\{{#1}\right\}}%figure bracets


\newcommand{\mb}{\left[\begin{array}}
	\newcommand{\me}{\end{array}\right]}

\newcommand{\nik}[1] {){#1} ð÷'(}

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%\newfont{\tax}{telaviv at 10pt}
\def\eps{\varepsilon}
%\def\R{{\mathbb R}}
%
\pagestyle{plain} \pagenumbering{arabic}

\renewcommand{\labelenumii}{\alph{enumii}.}
\def\eps{\varepsilon}
%\newcommand{\setR}{{\mathord{\mathbb R}}}
\newcommand{\Matrix}{{\mathord{\left[\begin{array}{r}
				x \\ x \\ x  \\
			\end{array}\right]}}}
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%

\newcommand{\widesim}[2][2.5]{
	\mathrel{\overset{#2}{\scalebox{#1}[1]{$\sim$}}}
}

\newcommand{\bse}[1]{\left(\begin{array}{#1|c}}
\def \ese {\end{array}\right)}

\def\F{\vec{\bf F}}
\def\hatu {\hat{\mathbf{u}}}
\def\n{\hat{\bf n}}
\def\r{\vec{\bf r}}
\def\i{\hat{\bf i}}
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\def\u{{\bf u}}
\def\v{{\bf v}}
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\def\c{{\bf c}}
\def\d{{\bf d}}
\def\f{{\bf f}}
\def\p{{\bf p}}
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% \vskip 5cm
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\def\part{\partial}
\def \bd {\left|\begin{matrix}}
\def \ed {\end{matrix}\right|}
\def \bbm {\begin{bmatrix}}
	\def \ebm {\end{bmatrix}}
\def \sp {\mathrm{sp}}
\def \tr {\mathrm{tr}}
\def \rp {\mathrm{Re}}
\def \ip {\mathrm{Im}}
\def \cis {\mathrm{cis}}
\def \bm {\left(\begin{matrix}}
	\def \em {\end{matrix}\right)}
\def \usualop {áéçñ ìôòåìåú äøâéìåú}
\def \pageoflines {\notes[24pt]{18}{\textwidth}}
\def \answer {\textbf{ôúøåï:}}

\def \S {\mathcal{S}}

%\newtheorem{theorem}{Theorem}
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\newtheorem{example}{\R{ãåâîä}}
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		\fontfamily{david}\selectfont \textbf{ùàìä}}}
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		\fontfamily{david}\selectfont \textbf{ôúøåï}}}
%\newtheorem{remark}[theorem]{Remark}

%\newcommand{{ ùàìä}}{\noindent{\bf \R{ .ùàìä}}}
%------------------------------------------------------------------------





\begin{document}
%\fontfamily{david}\selectfont



\begin{center}
{\textbf{ \Large{
 îèìä îñ' 5 àìâáøä 1îç'  \L{11102} \\
\quad \\
ìà ìäâùä
}}
\\}

\end{center}
%
%\textbf{äòøä:}
%áîèìä æå îåúø ìäùúîù áëì èòðä ùäåëéçå áîèìåú ÷åãîåú.



\begin{enumerate}

\item
éäé 
$(V, \setF)$
îøçá å÷èåøé ëàùø 
$\dim V=n$
å 
$B=\{\v_1, \ldots, \v_n\}$ 
áñéñ ì $V$.
äåëéçå àå äôøéëå àú äèòðåú äáàåú. 
\begin{enumerate}
\item 
ìëì å÷èåø 
$\y\in V$ 
ä÷áåöä 
$\{\y+\v_1, \ldots, \y+\v_n\}$ 
áñéñ ì $V$.

%\answer 
%
%äèòðä àéðä ðëåðä. ð÷ç 
%$\y=-\v_1$ 
%åàæ ä÷áåöä 
%\[\{\y+\v_1, \ldots, \y+\v_n\}=\{-\v_1+\v_1, \ldots, -\v_1+\v_n\}=\{\0, \ldots, -\v_1+\v_n\}\]
%îëéìä àú å÷èåø äàôñ ìëï äéà àéðä áú"ì )ëîåáï äéà âí ìà áñéñ(.

\item 
ìëì ñ÷ìøéí 
$\alpha_1, \ldots, \alpha_n\neq 0$
ä÷áåöä
$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$	
áñéñ ì 
$V$.

%\answer
%
%ä÷áåöä
%$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$	
%îú÷áìú îäáñéñ $B$ òì éãé ñãøä ùì ôòåìåú àìîðèøéåú. ìôé èòðä, ôòåìåú àìîðèøéåú ìà îùðåú úìåú ìéðàøéú àå ôøéùä ùì ÷áåöä. ëìï
%$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$
% âí áñéñ ì $V$.

\item 
áñòéó äæä áìáã
$V=\setR^n$
îøçá å÷èåøé îòì äùãä 
$\setR$ 
\usualop\ 
å 
${B=\{\v_1, \ldots, \v_n\}}$
áñéñ ì 
$\setR^n$.
òáåø
$A\in \setR^{n\times n}$ 
îèøéöä îñãø 
$n\times n$
ä÷áåöä 
$\{A\v_1, \ldots, A\v_n\}$ 
áñéñ ì $V$ àí åø÷ àí 
$A$ 
äôéëä.


%\answer
%
%ðæëéø ëé 
%$A$ 
%äôéëä àí åø÷ àí 
%$\Nu(A)=\{\0\}$.
%
%\textbf{ëéååï 1:}
%ððéç ëé $A$ äôéëä.
%éäéå 
%$\alpha_1, \ldots, \alpha_n\in \setR $
%ëê ù
%\begin{align*}
%\alpha_1 A\v_1+\cdots +\alpha_n A\v_n=\0.
%\end{align*}
%ðåáò ëé 
%\[A(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=\0.\]
%îëéååï ù 
%$A$ 
%äôéëä ðåëì ìäëôéì áùðé äàâôéí áîèøéöä äåôëéú 
%$A^{-1}$
%åð÷áì
%\[A^{-1}A(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=I(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=\alpha_1 \v_1+\cdots +\alpha_n \v_n=\0\]
%àáì $B$ áú"ì )ëé äåà áñéñ( ìëï
%$\alpha_1=\cdots=\alpha_n=0$.
%ðåáò ëé 
%$A\v_1,\ldots, A\v_n$
%áú"ì àáì 
%$\dim V=n$ 
%ìëï 
%$A\v_1,\ldots, A\v_n$
%îäååéí áñéñ ì 
%$V$.
%
%\textbf{ëéååï 2:}
%
%ððéç ù 
%$A\v_1,\ldots, A\v_n$
%îäååéí áñéñ ì 
%$\setR^n$.
%òì îðú ìäøàåú ù $A$ äôéëä, ðøàä ù
%$\Nu(A)=\{\0\}$.
%ìùí ëê, éäé 
%$\x\in \Nu(A)$.
%îëéååï 
%$B$
%áñéñ ì 
%$\setR^n$, 
%äéà ôåøùú àú 
%$\setR^n$.
%ìëï ÷ééîéí 
%$\beta_1, \ldots, \beta_n\in \setR$ 
%ëê ù 
%$\beta_1\v_1+\cdots+\beta_n \v_n=\x$.
%ðëôéì á 
%$A$ 
%åð÷áì 
%\begin{align*}
%&A(\beta_1\v_1+\cdots+\beta_n \v_n)=A\x=\0\\
%\implies & \beta_1A\v_1+\cdots+\beta_n A\v_n=\0.
%\end{align*}
%îëéååï ù 
%$\{A\v_1,\ldots, A\v_n\}$
%áñéñ ì 
%$\setR^n$ 
%äéà áú"ì. ðåáò ëé 
%$\beta_1=\cdots=\beta_n=0$. 
%îú÷áì ù 
%$\x=\beta_1\v_1+\cdots+\beta_n\v_n=\0$.
%ðåáò ëé 
%$\Nu(A)=\{\0\}$ 
%ìëï 
%$A$ 
%äôéëä.

\item 
éäéå 
$\a_1=\bm a_{11}\\ 0\\0\\0 \em, \a_2=\bm a_{12}\\ a_{22}\\0\\0 \em, \a_3=\bm a_{13}\\ a_{23}\\a_{33}\\0 \em, \a_4=\bm a_{14}\\ a_{24}\\a_{34}\\a_{44} \em$
å÷èåøéí á 
$\setR^4$ 
ëàùø 
$a_{11}, a_{22}, a_{33}, a_{44}\neq	 0$. 
äåëéçå ùäåå÷èåøéí îäååéí áñéñ ì 
$\setR^4$.

%\answer
%
%ðñãø àú äåå÷èåøéí äðúåðéí áîèøéöä:
%\[A=(\a_1,\a_2,\a_3,\a_4)=
%\bm a_{11}&a_{12}&a_{13}&a_{14}\\
% 0&a_{22}&a_{23}&a_{24}\\
% 0&0&a_{33}&a_{34}\\
% 0&0&0&a_{44} \em.
%\]
%ëòú, 
%$\r(A)=4$
%ëé äî÷ãîéí 
%$a_{11}, a_{22}, a_{33}, a_{44}$ 
%ùåðéí îàôñ ìëï ëì àçã îåáéì åäîèøéöä îãåøâú. ìôé èòðä, ðåáò ù 
%$A\in \setR^{4\times 4}$
% äôéëä.
\end{enumerate}


	
\item 
éäé 
$(\setR^3, \setR)$ 
îøçá å÷èåøé òí äôòåìåú äøâéìåú åéäé 
$V=\{f\in \setR[x]\mid \deg f\leq 2\}$ 
îøçá äôåìéðåîéí îòì 
$\setR$ 
îîòìä ìëì äéåúø 
$2$.
ðâãéø ôåð÷öéä 
$T:\setR^3\to V$ 
òì éãé 	
$T\bm a\\b\\c \em=(a+c)x^2+(a-b)x+(b+c)$
ìëì 
$\bm a\\b\\c\em\in \setR^3$.

\begin{enumerate}
\item	
äåëéçå ù $T$ äòú÷ä ìéðàøéú.

%\answer
%
%éäéå
%$\u_1=\bm a_1\\b_1\\c_1\em, \u_2=\bm a_2\\b_2\\c_2\em\in \setR^3$
%åéäé 
%$\alpha\in \setR$. 
%àæé îú÷ééîéí 
%\begin{align*}T(\u_1+\u_2)&=T\bm a_1+a_2\\b_1+b_2\\c_1+c_2\em\\
%	&=(a_1+a_2+c_1+c_2)x^2+(a_1+a_2-b_1-b_2)x+(b_1+b_2+c_1+c_2)\\
%	&=(a_1+    c_1    )x^2+(a_1    -b_1    )x+b_1    +c_1+(a_2+    c_2    )x^2+(a_2    -b_2    )x+b_2    +c_2\\
%	&=T(\u_1)+T(\u_2)
%	\end{align*}
%	å 
%\begin{align*}T(\alpha \u_1)&=(\alpha a_1+   \alpha c_1    )x^2+(\alpha a_1    -\alpha b_1    )x+(\alpha b_1    +\alpha c_1    )\\
%	&=\alpha \Big(a_1+    c_1    )x^2+(a_1    -b_1    )x+b_1    +c_1\Big)\\
%	&=\alpha T(\u_1).
%\end{align*}
%ðåáò ëé 
%$T$ 
%äòú÷ä ìéðàøéú.

\item 
äøàå ëé 
$x^2+1\in \ip(T)$
åâí 
$-x+1\in \ip(T)$.

%\answer 
%
%ð÷ç 
%$a=b=1, c=0$ 
%åð÷áì 
%$T\bm 1\\1\\0 \em=(1+0)x^2+(1-1)x+1+0=x^2+1$ 
%ìëï 
%$x^2+1\in \ip(T)$.
%
%ð÷ç 
%$a=c=0, b=1$ 
%åð÷áì 
%$T\bm 0\\0\\1 \em=(0+0)x^2+(0-1)x+1+0=-x+1$ 
%ìëï 
%$-x+1\in \ip(T)$.


\item 
äåëéçå ëé 
$\dim (\ip(T))\geq 2$.

%\answer 
%
%áñòéó ä÷åãí øàéðå ëé 
%$x^2+1, -x+1\in \ip(T)$. 
%áøåø ùäí áú"ì ëé äîòìåú ùåðåú.
%îöã ùðé, 
%$\sp\{x^2+1, -x+1\}\subseteq \ip(T)$.
% ðåáò ëé 
%\[\dim (\ip(T))\geq \dim (\sp\{ x^2+1, -x+1 \})=2.\]

\item 
äøàå ëé 
$\bm 1\\1\\-1 \em\in \ker(T)$.


%\answer
%
%ðçùá 
%\[T\bm 1\\1\\-1 \em=(1-1)x^2+(1-1)x+1-1=0\]
%ìëï 
%$\bm 1\\1\\-1\em\in \ker T$.

\item 
äåëéçå ëé 
$\dim (\ker(T))\geq 1$.


%\answer
%
%ðåáò îäñòéó ä÷åãí ëé 
%$\sp\left\{ \bm 1\\1\\-1 \em \right\}\subseteq \ker T$
%ìëï 
%\[\dim(\ker T)\geq \dim \left(\sp\left\{ \bm 1\\1\\-1 \em \right\}\right)=1.\]

\item 
çùáå àú 
$\dim (\ker (T))$ 
åàú 
$\dim (\ip(T))$.


%\answer
%
%ìôé îùôè îú÷ééí 
%$\dim (\ker (T))+\dim (\ip(T))=\dim(\setR^3)=3$. 
%ìôé äñòéôéí ÷åãîéí 
%$\dim (\ker (T))\geq 1$ 
%å 
%$\dim (\ip(T))\geq 2$
%ìëï áäëøç 
%$\dim (\ker (T))= 1$ 
%å 
%$\dim (\ip(T))= 2$.

%\item 
%éäé 
%$B=\{x^2, x, 1\}$
%äáñéñ äñèðãøèé ùì 
%$V$.
%úäé  
%$A\in \setR^{3\times 3}$
%îèøéöä äî÷ééîú 
%$[T(\u)]_B=A\u$ 
%ìëì 
%$\u\in \setR^3$.
%\begin{enumerate}
%	\item 
%	îöàå àú 
%	$A$ 
%	áîôåøù.
% 
%%\answer
%%
%%ðëúåá 
%%$\u=\bm a\\b\\c \em$
%%åðçùá 
%%\[[T\u]_B=[(a+c)x^2+(a-b)x+(b+c)]_B=\bm a+c\\ a-b\\ b+c \em.\]
%%ðãøù ìîöåà îèøéöä 
%%$A$ 
%%ëê ù 
%%$A\u=A\bm a \\ b \\ c \em=\bm a+c\\ a-b\\ b+c \em$.
%%ðéúï ì÷çú 
%%\[A=\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\]
%%ëé àæ 
%%\[A\bm a \\ b \\ c \em=\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\bm a \\ b \\ c \em=\bm a+c\\ a-b\\ b+c \em.\]
%
%\item
%äøàå ëé 
%$\Nu(A)=\sp\left\{ \bm 1\\1\\-1 \em\right\}$.
%
%%\answer
%%
%%
%%ëòú, 
%%$\Nu(A)$ 
%%äåà àåñó äôúøåðåú ìîî"ì 
%%$A\x=\0$. 
%%ðôúåø: 
%%\begin{align*}
%%(A|\0)=\bse{ccc} 
%%1&0&1&0\\ 1&-1&0&0\\ 0&1&1&0 \ese &
%%\widesim{R_2\to R_2-R_1} 
%%\bse{ccc} 
%%1&0&1&0\\ 0&-1&-1&0\\ 0&1&1&0 \ese\\
%%&\widesim{R_3\to R_3+R_2} 
%%\bse{ccc} 
%%1&0&1&0\\ 0&-1&-1&0\\ 0&0&0&0 \ese
%%\widesim{R_2\to -R_2} 
%%\bse{ccc} 
%%1&0&1&0\\ 0&1&1&0\\ 0&0&0&0 \ese.
%%\end{align*}
%%
%%ðëúåá 
%%$c=t$ 
%%åàæ ð÷áì 
%%$a=b=-t$ 
%%ìëï éù ôúøåï ëììé 
%%$\bm a\\ b\\ c\em=\left\{ t\bm -1\\-1\\1 \em\mid t\in \setR \right\}$
%%ìëï 
%%\[\Nu(A)=\sp \left\{\bm -1\\-1\\1 \em\right\}=\sp \left\{\bm 1\\1\\-1 \em\right\}.\]
%
%\item 
%îöàå áñéñ ì 
%$\C(A)$.
%
%%\answer
%%
%%ìôé ääâãøä ùì îøçá òîåãåú
%%$\C(A)=\sp\left\{
%%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em, \bm 1\\0\\1 \em
%%\right\}.$
%%ëáø øàéðå ëé 
%%$\bm 1\\ 1\\ -1\em\in \Nu(A)$ 
%%ìëï 
%%\[\bm 0\\0\\0 \em=\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\bm 1\\ 1\\ -1\em=1\bm 1\\1\\0\em+1\bm 0\\ -1\\ 1 \em+(-1)\bm 1\\0\\1 \em.\]
%%ìëï ðåëì ìëúåá àú äòîåãä äàçøåðä ëöéøåó ìéðàøé ùì äòîåãåú äàçøåú: 
%%\[\bm 1\\0\\1 \em=\bm 1\\1\\0\em+\bm 0\\ -1\\ 1 \em.\]
%%ðåáò ëé 
%%$\C(A)=\sp\left\{
%%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em, \bm 1\\0\\1 \em
%%\right\}=\sp\left\{
%%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em
%%\right\}.$
%%äåå÷èåøéí 
%%$\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em$ 
%%äí áú"ì ëé äí àéðí î÷áéìéí ìëï äí îäååéí áñéñ ì 
%%$\C(A)$. 

%\end{enumerate}


\end{enumerate}


	\item 
	éäé 
	$\setF$ 
	ùãä. 
		\begin{enumerate}
		\item 
			äåëéçå ùàí 
			$A, B\in \setF^{n\times n}$ 
îèøéöåú ñéîèøéåú àæé
$AB$ 
ñéîèøéú àí åø÷ àí 
$AB=BA$.
		 

%\answer 
%
%ðúåï ù 
%$A, B$ 
%ñéîèøéåú ìëï 
%$A^T=A, B^T=B$. 
%ëòú, 
%$(AB)^T=B^TA^T=BA$. 
%ðéúï ìäñé÷ ëé 
%$AB$
%ñéîèøéú àí åø÷ àí 
%$(AB)^T=AB$ 
%àí åø÷ àí 
%$BA=AB$.		
		
		\item 
		äåëéçå ùàí 
		$A, B\in \setR^{3\times 3}$
		îèøéöåú ñéîèøéåú 
		ëê ù 
		$AB=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$
		àæé âí 
		$BA=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$.

%\answer
%
%äîèøéöä
%$AB=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$
%ñéîèøéú ëé äéà àìëñåðéú. ìôé äñòéó ä÷åãí ðåáò ëé 
%$BA=AB=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$.

\item 
úðå ãåâîä ìîèøéöåú 
$A, B\in \setR^{2\times 2}$
)ìà áäëøç ñéîèøéåú( ëê ù 
$AB$
ñéîèøéú å
${AB\neq BA}$.		

%\answer 
%
%ð÷ç 
%$A=\bm 1&1\\0&1 \em, B=\bm 1&0\\ 1&1 \em$.
%àæ
%$AB=\bm 1&1\\0&1 \em \bm 1&0\\ 1&1 \em=\bm 2&1\\1&1 \em$
%ñéîèøéú å 
%$BA=\bm 1&0\\ 1&1 \em \bm 1&1\\0&1 \em = \bm 1&1\\1&2 \em \neq AB$.

	\end{enumerate}

	\item \label{inverse-of-products}
	éäé 
	$\setF$
	ùãä åúäééðä 
	$A, B\in \setF^{n\times n}$
	îèøéöåú äôéëåú. 
	\begin{enumerate}
		\item 
		äåëéçå ù 
		$AB$ 
		äôéëä.

%\answer
%
%ìôé îùôè 
%$\det (AB)=\det A\det B$. 
%îëéååï ù 
%$A, B$ 
%äôéëåú îú÷ééí 
%$\det A, \det B\neq 0$. 
%ðåáò ëé 
%$\det (AB)\neq 0$ 
%ìëï 
%$AB$ 
%äôéëä.
		\item 
		äåëéçå ëé 
		$(AB)^{-1}=B^{-1}A^{-1}$.
		
%\answer		
%		
%òì îðú ìäøàåú ëé 
%$B^{-1}A^{-1}$ 
%ääåôëéú ùì $AB$ àðå ðçùá: 
%\[ (AB)(B^{-1}A^{-1})= A(BB^{-1})A^{-1}=AIA^{-1}=AA^{-1}=I. \]		
%ìôé îùôè, îú÷ééí âí 
%$(B^{-1}A^{-1})(AB)=I$
%ìëï 
%$B^{-1}A^{-1}$
%àëï ääåôëéú ùì 
%$AB$.

\item 

àí 
$C\in \setF^{n\times n}$
âí äôéëä, îöàå ðåñçä
òáåø 
$(ABC)^{-1}$
ùáä îåôéòéí äîèøéöåú 
$A^{-1}, B^{-1}, C^{-1}$.

%\answer 
%
%ìôé äñòéó ä÷åãí îú÷ééí 
%$(ABC)^{-1}=((AB)C)^{-1}=C^{-1}(AB)^{-1}=C^{-1}B^{-1}A^{-1}$.
%		
		\item 		
		äåëéçå ù 
		$A^2$ 
		äôéëä.
		
%\answer 
%
%ðåáò éùéøåú îñòéó à' ëàùø 
%$B=A$
%ëé àæ
%$AB=A^2$.		
		
		\item
		äåëéçå ù
		$(A^2)^{-1}=(A^{-1})^2$.
		
%\answer	
%		
%		
%		ð÷ç 
%		$B=A$ 
%		åàæ ìôé ñòéó ùì äùàìä äðåëçéú á' ð÷áì
%		\[(A^2)^{-1}=(AA)^{-1}=(A^{-1})(A^{-1})=(A^{-1})^2.\]

		\item 
		àí 
		\[A=\bm -11&3&-4\\ -18&5&-7\\ 8&-2&3 \em, B=\bm 1&-1&1\\1&2&5\\ -2&4&1 \em\]
		çùáå àú 
		$AB$ 
		åàú 
		$B^{-1}A^{-1}$.
	îåîìõ ìäéòæø áñòéó á' áî÷åí ìçùá àú 
		$A^{-1}$ 
	åàú 
		$B^{-1}$ 
		áðôøã.
		
%\answer		
%		
%\[AB=\bm 0&1&0\\1&0&0\\0&0&1 \em.\]		
%ìôé ñòéó á' îú÷ééí 
%$B^{-1}A^{-1}=(AB)^{-1}=\bm 0&1&0\\1&0&0\\0&0&1 \em^{-1}$.	
%ðçùá àú ääåôëéú:
%\begin{align*}
%	\left(
%	\begin{array}{ccc|ccc}
%		0&1&0&1&0&0\\
%		1&0&0&0&1&0\\
%		0&0&1&0&0&1
%	\end{array} 	
%	\right) 	
%	&{\widesim{R_1\leftrightarrow R_2}}	
%	\left(
%	\begin{array}{ccc|ccc}
%1&0&0&0&1&0\\
%0&1&0&1&0&0\\
%0&0&1&0&0&1
%	\end{array} 	
%	\right).
%\end{align*}
%	ðåáò ëé 
%	$B^{-1}A^{-1}=\bm 0&1&0\\1&0&0\\0&0&1 \em^{-1}=\bm 0&1&0\\1&0&0\\0&0&1 \em$.
	
	\end{enumerate}	
	
	
	\item \label{eigen-of-inverse}
	éäé 
	$\setF$
	ùãä åúäé 
	$A\in \setF^{n\times n}$
	îèøéöä äôéëä. 
\begin{enumerate}
\item 
äåëéçå ùëì òøê òöîé ùì 
$A$ 
ùåðä îàôñ.

%\answer
%
%ðúåï ù 
%$A$ 
%äôéëä. ìëï, ìôé èòðä, 
%$\Nu(A)=\{\0\}$. 
%ðåáò ù 
%$A\x\neq \0$ 
%ìëì 
%$\x\neq \0$.
%ìëï ìà éúëï ù 
%$A\x=0\x$ 
%òáåø å÷èåø 
%$\x\neq \0$.
%ðåáò ù 
%$0$ 
%àéðå òøê òöîé ùì 
%$A$.



\item 
äåëéçå ùàí 
$\x$ 
å÷èåø òöîé ùì 
$A$ 
òí òøê òöîé 
$\lambda$ 
àæé 
$\x$ 
å÷èåø òöîé ùì 
$A^{-1}$
òí òøê òöîé 
$\frac{1}{\lambda}$.
ëàï 
$A^{-1}$ 
îñîï àú äîèøéöä ääåôëéú ùì $A$.

%\answer
%
%ìôé äðúåï îú÷ééí 
%$A\x=\lambda \x$. 
%ðëôéì àú ùðé äàâôéí á 
%$A^{-1}$ 
%åð÷áì 
%\[A^{-1}A\x=A^{-1}(\lambda \x)=\lambda A^{-1}\x.\]
%ëòú, ìôé ääâãøä ùì îèøéöä äåôëéú îú÷ééí 
%$A^{-1}A=I$ 
%ëàùø 
%$I$ 
%îèøéöú äéçéãä îñãø 
%$n\times n$.
%ðåáò ù 
%$A^{-1}A\x=I\x=\x$ 
%ìëï 
%$\x=\lambda A^{-1}\x$.
%ðéúï ìäñé÷ ëé 
%$A^{-1}\x=\frac{1}{\lambda}\x$
%ìëï 
%$\x$ 
%å÷èåø òöîé ùì 
%$A^{-1}$ 
%òí òøê òöîé 
%$\frac{1}{\lambda}$.

\item 
äåëéçå ùàí 
$\x$ 
å÷èåø òöîé ùì 
$A$ 
òí òøê òöîé 
$\lambda$ 
àæé 
$\x$ 
å÷èåø òöîé ùì 
$A^2$
òí òøê òöîé 
                    $\lambda^2$.

%\answer
%
%ìôé äðúåï 
%$A\x=\lambda\x$.
%ëòú,
%$A^2\x=AA\x=A\lambda \x=\lambda A\x=\lambda (\lambda \x)=\lambda^2\x$
%ìëï 
%$\x$ 
%å÷èåø òöîé ùì 
%$A^2$ 
%òí òøê òöîé 
%$\lambda^2$.


\end{enumerate}

\item 

	éäé 
	$\setF$
	ùãä åúäé 
	$A\in \setF^{n\times n}$
	îèøéöä äôéëä. 

\begin{enumerate}
	\item 
äåëéçå ëé 
$(A^T)^{-1}=(A^{-1})^T$.	


%\answer
%
%ëãé ìáãå÷ äàí 
%$(A^{-1})^T$
%ääåôëéú ùì 
%$A^T$,
%ðçùá àú 
%$A^T(A^{-1})^T$. 
%ëòú, 
%\begin{align*}
%A^T(A^{-1})^T&=(A^{-1}A)^T\\
%&=I^T=I.
%\end{align*}
%ìôé îùôè ðåáò ùâí 
%$(A^{-1})^TA^T=I$.
%ìëï àëï 
%$(A^{-1})^T$
%ääåôëéú ùì 
%$A^T$.

	\item 
äåëéçå ùàí 
$A$ 
ñéîèøéú àæé âí 
$A^{-1}$
ñéîèøéú.

%\answer
%
%àí $A$ ñéîèøéú àæ 
%$A^T=A$. 
%ìôé ñòéó ä÷åãí ð÷áì 
%\[(A^{-1})^T=(A^T)^{-1}=A^{-1}\]
%ìëï 
%$A^{-1}$
%ñéîèøéú.
\item 
àí 
\[A=\bm 3&0&5\\0&2&1\\5&1&9 \em\]
\begin{enumerate}
	\item
çùáå àú 
$A^{-1}$.

%\answer
%
%òì îðú ìçùá àú ääåëôéú ùì $A$ ðãøâ:
%
%\begin{align*}
%	\left(
%	\begin{array}{ccc|ccc}
%		3&0&5&1&0&0\\
%		0&2&1&0&1&0\\
%		5&1&9&0&0&1
%	\end{array} 	
%	\right) 	
%	&{\widesim{R_3\to 3R_3-5R_1}}	
%	\left(
%	\begin{array}{ccc|ccc}
%		3&0&5&1&0&0\\
%		0&2&1&0&1&0\\
%		0&3&2&-5&0&3
%	\end{array} 	
%	\right)\\[0.5em]
%	&{\widesim{R_3\to 2R_3-3R_2}}	
%	\left(
%	\begin{array}{ccc|ccc}
%		3&0&5&1&0&0\\
%		0&2&1&0&1&0\\
%		0&0&1&-10&-3&6
%	\end{array} 	
%	\right)\\[0.5em]
%		&{\widesim{R_1\to R_1-5R_2}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			3&0&0&51&15&-30\\
%			0&2&1&0&1&0\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\[0.5em]
%		&{\widesim{R_2\to R_2-R_3}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			3&0&0&51&15&-30\\
%			0&2&0&10&4&-6\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\[0.5em]
%		&{\widesim{R_1\to \frac{1}{3}R_1}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			1&0&0&17&5&-10\\
%			0&2&0&10&4&-6\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\[0.5em]				
%		&{\widesim{R_2\to \frac{1}{2}R_2}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			1&0&0&17&5&-10\\
%			0&1&0&5&2&-3\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\						
%\end{align*}
%ìëï 
%$A^{-1}=\bm
%17&5&-10\\
%5&2&-3\\
%-10&-3&6
%\em$.

\item 
çùáå àú 
$(A^2)^{-1}$.

%\answer
%
%ìôé ùàìä 
%\ref{inverse-of-products}
% ñòéó ä' îú÷ééí 
%$(A^2)^{-1}=(A^{-1})^2$.
%ìôé äñòéó ä÷åãí ð÷áì 
%\begin{align*}
%(A^2)^{-1}&=(A^{-1})^2\\
%&=A^{-1}A^{-1}\\
%&=\bm
%17&5&-10\\
%5&2&-3\\
%-10&-3&6
%\em\bm
%17&5&-10\\
%5&2&-3\\
%-10&-3&6
%\em\\
%&=\bm
%414&125&-245\\
%125&38&-74\\
%-245&-74&145
%\em
%\end{align*}

\end{enumerate}
\end{enumerate}



\textbf{úæëåøú:}
òáåø îèøéöä äôéëä
$\bm p&q\\ r&s \em$
 îñãø 
$2\times 2$
äîèøéöä ääåôëéú ðúåðä òì éãé 
\[\bm p&q\\ r&s \em^{-1}=\frac{1}{ps-qr}\bm s&-q\\ -r&p \em.\]

\item 

éäéå
	$a, b\in \setR$ 
	ëê ùîú÷ééí 
	$a^2-b^2=1$.
	ðâãéø îèøéöåú 
	\[C=\bm a&b\\ b&a \em, D=\bm a&b\\ a&b \em \in \setR^{2\times 2}.\]

\begin{enumerate}
	\item 
	äøàå ù $D$ àéðä äôéëä.

%\answer 
%
%\textbf{ùéèä 1:}	
%$\det D=ab-ba=0$ 
%ìëï $D$ àéðä äôéëä.
%
%\textbf{ùéèä 2:}
%äòîåãåú ùì $D$ î÷áéìåú ìëï äï úìåéåú ìéðàøéú. ðåáò ëé 
%$D$ 
%àéðä äôéëä.
%
%
%\textbf{ùéèä 3:}
%äùåøåú ùì $D$ æäåú ìëï 
%$\r (D)<2$ 
%)áòöí 
%$\r (D)=1$ 
%ëé 
%$a^2-b^2=1$ 
%ìëï 
%$(a, b)\neq (0,0)$.(
%ðåáò ëé 
%$D$ 
%àéðä äôéëä.
%
%\textbf{ùéèä 4:}
%$D\bm b\\-a \em=\bm 0\\0 \em$
%ìëï 
%$\bm b\\ -a\em\in \Nu(D)$.
%àáì 
%$\bm b\\-a \em\neq \bm 0\\0 \em$
%ëé 
%$a^2-b^2=1$.
%ðåáò ëé 
%$\Nu(D)\neq \{\0\}$ 
%ìëï 
%$D$ 
%àéðä äôéëä.
	
	\item 
	äøàå ëé 
	$C$ 
	äôéëä åîöàå àú äîèøéöä ääåôëéú 
	$C^{-1}$.
	
%\answer	
%	
%ðçùá: 
%$\det C=a^2-b^2=1\neq 0$ 
%ìëï 
%$C$ 
%äôéëä. ìôé èòðä ääåôëéú ðúåðä òì éãé 
%$C^{-1}=\frac{1}{\det C}\bm a&-b\\ -b&a \em=\bm a&-b\\ -b&a \em$.
	
	\item 
	îöàå àú ëì äòøëéí äòöîééí ùì 
	$C$
)éù ìëúåá àú äò"ò áàîöòåú äôøîèøéí
$a, b$.(	
	òáåø ëì òøê òöîé îöàå å÷èåø òöîé îúàéí.
	
%\textbf{ùéèä 1:}
%ìîèøéöä 
%$C$ 
%éù äøáä ñéîèøéä. áôøè, ñëåí äî÷ãîéí áëì ùåøä ùååä. ðåáò ù 
%$\bm 1\\ 1\em$ 
%å÷èåø òöîé ùì 
%$C$ 
%àí òøê òöîé 
%$a+b$:
%
%\[C\bm 1\\ 1\em=\bm a&b\\ b&a \em\bm 1\\1 \em=\bm a+b\\ a+b \em=(a+b)\bm 1\\1 \em.\]	
%áàåôï ãåîä, äôøù äî÷ãîéí áëì ùåøä ãåîä ìëï ðáãå÷ àú äåå÷èåø 
%$\bm 1\\ -1\em$:
%
%\[C\bm 1\\ -1\em=\bm a&b\\ b&a \em\bm 1\\-1 \em=\bm a-b\\ b-a \em=(a-b)\bm 1\\-1 \em.\]
%ìîèøéöä îñãø 
%$2\times 2$ 
%éù ìëì äéåúø ùðé ùåøùéí ìôåìéðåí äàåôééðé ìëï éù ìëì äéåúø ùðé òøëéí òöîééí. ðéúï ìäñé÷ ùîöàðå àú ëì äòøëéí äòöîééí ùì $C$.
%\textbf{ùéèä 2:}
%ðçùá àú äôåìéðåí äàåôééðé ùì $C$:
%\begin{align*}
%p_C(\lambda)&=\bd a-\lambda&b\\ b&a-\lambda \ed\\
%&=(a-\lambda)^2-b^2\\
%&=\lambda^2-2a\lambda+a^2-b^2\\
%&=(\lambda-(a+b))(\lambda-(a-b))	
%\end{align*}	
%ìëï éù òøëéí òöîééí 
%$a+b,\ \ a-b$.
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$a+b$ 
%ðîöà å÷èåø ìà àôñ á 
%$\Nu(A-(a+b)I)$:
%\begin{align*}
%(A-(a+b)I|\0)&=\bse{cc} -b&b&0\\ b&-b&0 \ese{\widesim{R_2\to R_2+R_1}}\bse{cc} -b&b&0\\ 0&0&0 \ese
%\end{align*}	
%àæ ðéúï ì÷çú å"ò
%$\v_1=\bm1\\1 \em$.\\
%
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$a-b$ 
%ðîöà å÷èåø ìà àôñ á 
%$\Nu(A-(a-b)I)$:
%\begin{align*}
%	(A-(a-b)I|\0)&=\bse{cc} b&b&0\\ b&b&0 \ese{\widesim{R_2\to R_2-R_1}}\bse{cc} b&b&0\\ 0&0&0 \ese
%\end{align*}	
%àæ ðéúï ì÷çú å"ò
%$\v_2=\bm1\\-1 \em$.
		\item 
		îöàå àú ëì äòøëéí äòöîééí ùì 
		$D$
		)éù ìëúåá àú äò"ò áàîöòåú äôøîèøéí
		$a, b$.(
		òáåø ëì òøê òöîé îöàå å÷èåø òöîé îúàéí.
		
%\textbf{ùéèä 1:}
%ëîå áñòéó ä÷åãí, ìîèøéöä 
%$D$ 
%éù äøáä ñéîèøéä. áôøè, ñëåí äî÷ãîéí áëì ùåøä ùååä. ðåáò ù 
%$\bm 1\\ 1\em$ 
%å÷èåø òöîé ùì 
%$D$ 
%àí òøê òöîé 
%$a+b$:
%
%\[D\bm 1\\ 1\em=\bm a&b\\ a&b \em\bm 1\\1 \em=\bm a+b\\ a+b \em=(a+b)\bm 1\\1 \em.\]	
%ëòú, ðéúï ìàôñ àú ùúé äùåøåú àí ðëôéì áåå÷èåø
%$\bm b\\ -a\em$
%ìëï äåà å÷èåø òöîé òí òøê òöîé 
%$0$:
%
%\[D\bm b\\ -a\em=\bm a&b\\ a&b \em\bm b\\-a \em=\bm ab-ba\\ ab-ba \em=\bm 0\\0 \em=0\bm b\\-a \em.\]
%ìîèøéöä îñãø 
%$2\times 2$ 
%éù ìëì äéåúø ùðé ùåøùéí ìôåìéðåí äàåôééðé ìëï éù ìëì äéåúø ùðé òøëéí òöîééí. ðéúï ìäñé÷ ùîöàðå àú ëì äòøëéí äòöîééí ùì $D$.
%\textbf{ùéèä 2:}
%ðçùá àú äôåìéðåí äàåôééðé ùì $D$:
%\begin{align*}
%	p_D(\lambda)&=\bd a-\lambda&b\\ a&b-\lambda \ed\\
%	&=(a-\lambda)(b-\lambda)-ab\\
%	&=\lambda^2-(a+b)\lambda\\
%	&=\lambda(\lambda-(a+b))
%\end{align*}	
%ìëï éù òøëéí òöîééí 
%$a+b,\ \ 0$.
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$a+b$ 
%ðîöà å÷èåø ìà àôñ á 
%$\Nu(A-(a+b)I)$:
%\begin{align*}
%	(A-(a+b)I|\0)&=\bse{cc} -b&b&0\\ a&-a&0 \ese{\widesim{}}\bse{cc} 1&-1&0\\ 0&0&0 \ese
%\end{align*}	
%îëéååï ù 
%$a^2-b^2=1$ 
%ìëï 
%$a\neq 0$ 
%àå 
%$b\neq 0$ 
%åäùåøåú î÷áéìåú.
%àæ ðéúï ì÷çú å"ò
%$\v_1=\bm1\\1 \em$.\\
%
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$0$ 
%ðîöà å÷èåø ìà àôñ á 
%${\Nu(A-(0)I)}$:
%\begin{align*}
%	(A-(0)I|\0)=(A|\0)&=\bse{cc} a&b&0\\ a&b&0 \ese{\widesim{R_2\to R_2-R_1}}\bse{cc} a&b&0\\ 0&0&0 \ese
%\end{align*}	
%àæ ðéúï ì÷çú å"ò
%$\v_2=\bm b\\-a \em$
%åäåà àéðå å÷èåø äàôñ ëé 
%$a\neq 0$ 
%àå 
%$b\neq 0$ 
%ëôé ùëáø öééðå.		
\item 
	äøàå ëé éù ì 
	$C$ 
	å 
	$C^{-1}$ 
	àåúí òøëéí òöîééí.
	
%	\answer
%	
%	ìôé ñòéó á' ùì äùàìä äðåëçéú äîèøéöä 
%	$C$ 
%	äôéëä.
%	ìôé ùàìä 
%	\ref{eigen-of-inverse}
%	ñòéó á' ìëì òøê òöîé 
%	$\lambda$ 
%	ùì $C$ éù ì
%	$C^{-1}$ 
%	éùðí òøëéí òöîééí 
%	 $\frac{1}{\lambda}$.
%	 ìôé ñòéó â' ùì äùàìä äðåëçéú äòøëéí äòöîééí ùì $C$ äí 
%	 $a+b,\ a-b$. 
%	 ðåáò ùäòøëéí äòöîééí ùì 
%	 $C^{-1}$ 
%	 äí 
%	 $\frac{1}{a+b}, \ \frac{1}{a-b}$.
%	 ðúåï ëé 
%	 $1=a^2-b^2=(a+b)(a-b)$
%	 ìëï
%	 $\frac{1}{a+b}=a-b$
%	 å 
%	 $\frac{1}{a-b}=a+b$.
%	 ìëï ì 
%	 $C^{-1}$ 
%éù àåúí òøëéí òöîééí 
%	 $a+b, \ a-b$ 
%	 ëîå 
%	 $C$.
%	
	\item 
	úðå ãåâîä ìîèøéöä äôéëä $Y$ îñãø 
	$2\times 2$ 
	ëê ùäòøëéí òöîééí ùì 
	$Y$ 
	ùåðéí îäòøëéí äòöîééí ùì 
	$Y^{-1}$.
	
	
%	\answer
%	
%ð÷ç ìîùì
%$Y=\bm 2&0\\ 0&3 \em$
%åàæ 
%$Y^{-1}=\bm \frac{1}{2}&0\\ 0&\frac{1}{3} \em$.
%äòøëéí äòöîééí ùì 
%$Y$ 
%äí 
%$2, 3$
%)ðéúï ì÷çú ìîùì å÷èåøéí òöîééí 
%$\bm 1\\ 0\em, \bm 0\\ 1\em$.(
%áàåôï ãåîä, ÷ì ìøàåú ùäòøëéí äòöîééí ùì 
%$Y^{-1}$ 
%äí 
%$\frac{1}{2}, \frac{1}{3}$. 
%ìëï ì 
%$Y, Y^{-1}$
%éù òøëéí òöîééí ùåðéí.
	
\end{enumerate}

\item 
ðâãéø ÷áåöú îèøéöåú òì éãé 
\[\S=\left\{ \bm a&-b\\ b&a  \em\mid a,b\in \setR, a^2+b^2=1 \right\}.\]
\begin{enumerate}
	\item 
	äøàå ùëì îèøéöä á 
	$\S$ 
	äôéëä.
	
%	\answer 
%	
%àí 
%$A=\bm a&-b\\ b&a\em\in \S$ 
%àæ 
%$\det A=a^2-(-b^2)=a^2+b^2=1\neq 0$ 
%ìëï 
%$A$ 
%äôéëä.

	\item 
	äåëéçå ùìëì 
	$A\in \S$ 
	îú÷ééí 
	$A^{-1}\in \S$.
	
%\answer	
%	
%àí 
%$A=\bm a&-b\\ b&a\em\in \S$ 	
%	àæ ìôé èòðä îú÷ééí 
%\[A^{-1}=\frac{1}{\det A}\bm a&b\\ -b&a \em=\frac{1}{a^2+b^2}\bm a&b\\ -b&a \em=\frac{1}{1}\bm a&b\\ -b&a \em=\bm a&-(-b)\\ b&a \em\in \S\]
%)äúðàé 
%$a^2+b^2=1$
%òãééï îú÷ééí.(	
	
	\item 
	äåëéçå ùìëì 
	$A, B\in \S$
	îú÷ééí 
	$AB\in \S$.
	
%\answer
%
%úééðä 
%	$A=
%	\bm a&-b\\ b&a \em, B=
%	\bm c&-d\\ d&c \em.
%	$
%	
%	
%	ëòú, 
%	\begin{align*}
%	AB&=\bm a&-b\\ b&a \em
%	\bm c&-d\\ d&c \em=\bm ac-bd&-ad-bc \\bc+ad&ac-bd  \em	
%	\end{align*}
	
	\item 
	îöàå ëì äôúøåðåú ìîùååàä 
	$A^2=\bm 1&0\\ 0&1 \em$
	ëàùø 
	$A\in \S$.
	
%\answer 
%
%ð÷ç 
%$a=-1, b=0$ 
%åð÷áì 	
%	$A=\bm a&-b\\b&a \em=\bm -1&0\\0&-1 \em$.
%	îëàï ð÷áì 
%	\[A^2=\bm -1&0\\0&-1 \em^2=\bm -1&0\\0&-1 \em\bm -1&0\\0&-1 \em=\bm 1&0\\0&1 \em\]
%	
\end{enumerate}

%\item 
%
%\begin{enumerate}
%	\item 
%	\item 
%\end{enumerate}




%\item 
%
%\begin{enumerate}
%	\item 
%	\item 
%\end{enumerate}





\end{enumerate}



\textbf{äåøàåú:}

\begin{itemize}
	\item
îèìä æå àéðä ìäâùä.
	\item 
ôúøåðåú éôåøñîå ìëì äîàåçø á 11 áéåìé.

\end{itemize}













\end{document}

Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Date: Thu, 18 Jul 2019 14:31:44 +0300
> 
> I attach the file, once as a original file, once with the latex commands
> removed.

I don't understand what files you attached -- are those the files
presumably encoded in ISO8859-8, or after conversion to UTF-8?  What I
see are UTF-8 encoded files, but the characters in them are not Hebrew
characters, they are Latin characters.

Can you send the original files, before any conversion/mangling?

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

[-- Attachment #1.1: Type: text/plain, Size: 862 bytes --]

>>> "EZ" == Eli Zaretskii <eliz@gnu.org> writes:

>> From: Uwe Brauer <oub@mat.ucm.es>
>> Date: Thu, 18 Jul 2019 14:31:44 +0300
>> 
>> I attach the file, once as a original file, once with the latex commands
>> removed.

> I don't understand what files you attached -- are those the files
> presumably encoded in ISO8859-8, or after conversion to UTF-8?  What I
> see are UTF-8 encoded files, but the characters in them are not Hebrew
> characters, they are Latin characters.

These are the original files, for me GNU emacs present them as they were
latin-1 encoded, I attach a screen shot.
Please note the 1 in the left lower corner.

These symbols are  displayed in texstudio, texmaker and in kile as
hebrew if iso-8859-8 is enabled.


> Can you send the original files, before any conversion/mangling?

These are the original files.


[-- Attachment #1.2: emacs-hebrew8859-8.png --]
[-- Type: image/png, Size: 48101 bytes --]

[-- Attachment #2: smime.p7s --]
[-- Type: application/pkcs7-signature, Size: 5025 bytes --]

Re: problem with old hebrew latex files in iso 8859-8 coding

[Lars Ingebrigtsen]

Uwe Brauer <oub@mat.ucm.es> writes:

> These are the original files, for me GNU emacs present them as they were
> latin-1 encoded, I attach a screen shot.
> Please note the 1 in the left lower corner.

Gnus has encoded the file as utf-8:

Content-Type: text/plain; charset=utf-8
Content-Disposition: attachment; filename=document.txt
Content-Transfer-Encoding: base64

Try renaming the file to document.bin and resend it as octet-stream...

-- 
(domestic pets only, the antidote for overdose, milk.)
   bloggy blog: http://lars.ingebrigtsen.no

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

[-- Attachment #1.1: Type: text/plain, Size: 172 bytes --]


   > Uwe Brauer <oub@mat.ucm.es> writes:

   > Try renaming the file to document.bin and resend it as octet-stream...

Thanks for pointing it out! I will do that 


[-- Attachment #1.2: document.bin --]
[-- Type: application/octet-stream, Size: 105 bytes --]

éäé  

îøçá å÷èåøé ëàùø 

å 

áñéñ ì  
äåëéçå àå äôøéëå àú äèòðåú äáàåú. 


ìëì å÷èåø 

ä÷áåöä 

áñéñ ì 

[-- Attachment #1.3: junk.bin --]
[-- Type: application/octet-stream, Size: 25582 bytes --]

\documentclass[11pt]{article}
%\usepackage[8859-8]{inputenc}
\usepackage[english,hebrew]{babel}
\usepackage{color}
%\usepackage{hebcal}
\usepackage{culmus}
\usepackage{courier}
\usepackage{ulem}
%\usepackage[HE8,OT1]{fontenc}
%\usepackage{ccfonts}
%\usepackage{float}
% \usepackage{color}
\usepackage{theorem}
\usepackage{fancyhdr}
\usepackage{esvect}
\usepackage{graphicx}
\usepackage{epstopdf}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usetikzlibrary{calc}
%\usepackage{url}
%\usepackage{listings}
%\usepackage{a4wide}
%\renewcommand{\baselinestretch}{1.5}
%\usepackage{graphicx,epsf}

%\renewcommand{\labelenumi}{\alph{enumi}.}
\usepackage{amssymb,amsfonts,amsmath}
\usepackage{pgffor, ifthen}
\newcommand{\notes}[3][\empty]{%
	\noindent \vspace{10pt}\\
	\foreach \n in {1,...,#2}{%
		\ifthenelse{\equal{#1}{\empty}}
		{\rule{#3}{0.5pt}\\}
		{\rule{#3}{0.5pt}\vspace{#1}\\}
	}
}


\setlength{\textheight}{25cm}
% \setlength{\textwidth}{6.6in}
\setlength{\topmargin}{-0.8in}
% \setlength{\textwidth}{6.6in}
% \setlength{\textheight}{1.20\textheight}
% \setlength{\oddsidemargin}{-0.25in}
% \setlength{\evensidemargin}{-0.25in}


%% \newcounter{parnum}
%% \newcommand{\pg}{%
%%    \noindent\refstepcounter{parnum}%
%%     \makebox[\parindent][l]{\textbf{\arabic{parnum}.}}}
% Use a generous paragraph indent so numbers can be fit inside the
% indentation space.
\usepackage{a4wide}

\setlength{\parindent}{2em}

\setlength{\textheight}{24.5cm}
% \setlength{\textwidth}{6.6in}
\setlength{\topmargin}{-0.9in}
% \setlength{\textwidth}{6.6in}
% \setlength{\textheight}{1.20\textheight}
% \setlength{\oddsidemargin}{-0.25in}
% \setlength{\evensidemargin}{-0.25in}

\renewcommand*\rmdefault{david}

\newcommand{\setB}{{\mathord{\mathbb B}}}
\newcommand{\setC}{{\mathord{\mathbb C}}}
\newcommand{\setF}{{\mathord{\mathbb F}}}
\newcommand{\setN}{{\mathord{\mathbb N}}}
\newcommand{\setQ}{{\mathord{\mathbb Q}}}
\newcommand{\setR}{{\mathord{\mathbb R}}}
\newcommand{\setZ}{{\mathord{\mathbb Z}}}
\newcommand{\nk}[1] {\textcolor{red}{)#1 ð÷'(}}
\newcommand{\red}[1] {\textcolor{red}{#1}}
%
\newcommand{\br}[1]{\left[{#1}\right]} %square bracets
\newcommand{\brf}[1]{\left\{{#1}\right\}}%figure bracets


\newcommand{\mb}{\left[\begin{array}}
	\newcommand{\me}{\end{array}\right]}

\newcommand{\nik}[1] {){#1} ð÷'(}

\newcommand{\ooiint}{\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}}
%\newfont{\ta}{telaviv at 12pt}
%\newfont{\tax}{telaviv at 10pt}
\def\eps{\varepsilon}
%\def\R{{\mathbb R}}
%
\pagestyle{plain} \pagenumbering{arabic}

\renewcommand{\labelenumii}{\alph{enumii}.}
\def\eps{\varepsilon}
%\newcommand{\setR}{{\mathord{\mathbb R}}}
\newcommand{\Matrix}{{\mathord{\left[\begin{array}{r}
				x \\ x \\ x  \\
			\end{array}\right]}}}
%
\pagestyle{plain} \pagenumbering{arabic}
%

\newcommand{\widesim}[2][2.5]{
	\mathrel{\overset{#2}{\scalebox{#1}[1]{$\sim$}}}
}

\newcommand{\bse}[1]{\left(\begin{array}{#1|c}}
\def \ese {\end{array}\right)}

\def\F{\vec{\bf F}}
\def\hatu {\hat{\mathbf{u}}}
\def\n{\hat{\bf n}}
\def\r{\vec{\bf r}}
\def\i{\hat{\bf i}}
\def\j{\hat{\bf j}}
\def\k{\hat{\bf k}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\u{{\bf u}}
\def\v{{\bf v}}
\def\w{{\bf w}}
\def\z{{\bf z}}
\def\a{{\bf a}}
\def\b{{\bf b}}
\def\c{{\bf c}}
\def\d{{\bf d}}
\def\f{{\bf f}}
\def\p{{\bf p}}
\def\q{{\bf q}}
\def\e{{\bf e}}
\def\N{{\bf N}}
\def\0{{\bf 0}}
\def\C{{\rm  Col}}
\def\r{\rm  rank}
\def\-{\L{-}}

% \vskip 5cm
\def \image {\mathrm{Im}}
\def \ker {\mathrm{ker}}
\def\Nu{{\rm Nul}}
\def\l{\lambda}
\def\la{\langle}
\def\ra{\rangle}
\def\part{\partial}
\def \bd {\left|\begin{matrix}}
\def \ed {\end{matrix}\right|}
\def \bbm {\begin{bmatrix}}
	\def \ebm {\end{bmatrix}}
\def \sp {\mathrm{sp}}
\def \tr {\mathrm{tr}}
\def \rp {\mathrm{Re}}
\def \ip {\mathrm{Im}}
\def \cis {\mathrm{cis}}
\def \bm {\left(\begin{matrix}}
	\def \em {\end{matrix}\right)}
\def \usualop {áéçñ ìôòåìåú äøâéìåú}
\def \pageoflines {\notes[24pt]{18}{\textwidth}}
\def \answer {\textbf{ôúøåï:}}

\def \S {\mathcal{S}}

%\newtheorem{theorem}{Theorem}
\newtheorem{theorem}{\R{îùôè}}
\newtheorem{example}{\R{ãåâîä}}
%\newtheorem{question}{\R{ùàìä}}
\theorembodyfont{\fontfamily{david}\selectfont}\newtheorem{question}{\R{
		\fontfamily{david}\selectfont \textbf{ùàìä}}}
\theorembodyfont{\fontfamily{david}\selectfont}\newtheorem{solution}{\R{
		\fontfamily{david}\selectfont \textbf{ôúøåï}}}
%\newtheorem{remark}[theorem]{Remark}

%\newcommand{{ ùàìä}}{\noindent{\bf \R{ .ùàìä}}}
%------------------------------------------------------------------------





\begin{document}
%\fontfamily{david}\selectfont



\begin{center}
{\textbf{ \Large{
 îèìä îñ' 5 àìâáøä 1îç'  \L{11102} \\
\quad \\
ìà ìäâùä
}}
\\}

\end{center}
%
%\textbf{äòøä:}
%áîèìä æå îåúø ìäùúîù áëì èòðä ùäåëéçå áîèìåú ÷åãîåú.



\begin{enumerate}

\item
éäé 
$(V, \setF)$
îøçá å÷èåøé ëàùø 
$\dim V=n$
å 
$B=\{\v_1, \ldots, \v_n\}$ 
áñéñ ì $V$.
äåëéçå àå äôøéëå àú äèòðåú äáàåú. 
\begin{enumerate}
\item 
ìëì å÷èåø 
$\y\in V$ 
ä÷áåöä 
$\{\y+\v_1, \ldots, \y+\v_n\}$ 
áñéñ ì $V$.

%\answer 
%
%äèòðä àéðä ðëåðä. ð÷ç 
%$\y=-\v_1$ 
%åàæ ä÷áåöä 
%\[\{\y+\v_1, \ldots, \y+\v_n\}=\{-\v_1+\v_1, \ldots, -\v_1+\v_n\}=\{\0, \ldots, -\v_1+\v_n\}\]
%îëéìä àú å÷èåø äàôñ ìëï äéà àéðä áú"ì )ëîåáï äéà âí ìà áñéñ(.

\item 
ìëì ñ÷ìøéí 
$\alpha_1, \ldots, \alpha_n\neq 0$
ä÷áåöä
$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$	
áñéñ ì 
$V$.

%\answer
%
%ä÷áåöä
%$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$	
%îú÷áìú îäáñéñ $B$ òì éãé ñãøä ùì ôòåìåú àìîðèøéåú. ìôé èòðä, ôòåìåú àìîðèøéåú ìà îùðåú úìåú ìéðàøéú àå ôøéùä ùì ÷áåöä. ëìï
%$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$
% âí áñéñ ì $V$.

\item 
áñòéó äæä áìáã
$V=\setR^n$
îøçá å÷èåøé îòì äùãä 
$\setR$ 
\usualop\ 
å 
${B=\{\v_1, \ldots, \v_n\}}$
áñéñ ì 
$\setR^n$.
òáåø
$A\in \setR^{n\times n}$ 
îèøéöä îñãø 
$n\times n$
ä÷áåöä 
$\{A\v_1, \ldots, A\v_n\}$ 
áñéñ ì $V$ àí åø÷ àí 
$A$ 
äôéëä.


%\answer
%
%ðæëéø ëé 
%$A$ 
%äôéëä àí åø÷ àí 
%$\Nu(A)=\{\0\}$.
%
%\textbf{ëéååï 1:}
%ððéç ëé $A$ äôéëä.
%éäéå 
%$\alpha_1, \ldots, \alpha_n\in \setR $
%ëê ù
%\begin{align*}
%\alpha_1 A\v_1+\cdots +\alpha_n A\v_n=\0.
%\end{align*}
%ðåáò ëé 
%\[A(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=\0.\]
%îëéååï ù 
%$A$ 
%äôéëä ðåëì ìäëôéì áùðé äàâôéí áîèøéöä äåôëéú 
%$A^{-1}$
%åð÷áì
%\[A^{-1}A(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=I(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=\alpha_1 \v_1+\cdots +\alpha_n \v_n=\0\]
%àáì $B$ áú"ì )ëé äåà áñéñ( ìëï
%$\alpha_1=\cdots=\alpha_n=0$.
%ðåáò ëé 
%$A\v_1,\ldots, A\v_n$
%áú"ì àáì 
%$\dim V=n$ 
%ìëï 
%$A\v_1,\ldots, A\v_n$
%îäååéí áñéñ ì 
%$V$.
%
%\textbf{ëéååï 2:}
%
%ððéç ù 
%$A\v_1,\ldots, A\v_n$
%îäååéí áñéñ ì 
%$\setR^n$.
%òì îðú ìäøàåú ù $A$ äôéëä, ðøàä ù
%$\Nu(A)=\{\0\}$.
%ìùí ëê, éäé 
%$\x\in \Nu(A)$.
%îëéååï 
%$B$
%áñéñ ì 
%$\setR^n$, 
%äéà ôåøùú àú 
%$\setR^n$.
%ìëï ÷ééîéí 
%$\beta_1, \ldots, \beta_n\in \setR$ 
%ëê ù 
%$\beta_1\v_1+\cdots+\beta_n \v_n=\x$.
%ðëôéì á 
%$A$ 
%åð÷áì 
%\begin{align*}
%&A(\beta_1\v_1+\cdots+\beta_n \v_n)=A\x=\0\\
%\implies & \beta_1A\v_1+\cdots+\beta_n A\v_n=\0.
%\end{align*}
%îëéååï ù 
%$\{A\v_1,\ldots, A\v_n\}$
%áñéñ ì 
%$\setR^n$ 
%äéà áú"ì. ðåáò ëé 
%$\beta_1=\cdots=\beta_n=0$. 
%îú÷áì ù 
%$\x=\beta_1\v_1+\cdots+\beta_n\v_n=\0$.
%ðåáò ëé 
%$\Nu(A)=\{\0\}$ 
%ìëï 
%$A$ 
%äôéëä.

\item 
éäéå 
$\a_1=\bm a_{11}\\ 0\\0\\0 \em, \a_2=\bm a_{12}\\ a_{22}\\0\\0 \em, \a_3=\bm a_{13}\\ a_{23}\\a_{33}\\0 \em, \a_4=\bm a_{14}\\ a_{24}\\a_{34}\\a_{44} \em$
å÷èåøéí á 
$\setR^4$ 
ëàùø 
$a_{11}, a_{22}, a_{33}, a_{44}\neq	 0$. 
äåëéçå ùäåå÷èåøéí îäååéí áñéñ ì 
$\setR^4$.

%\answer
%
%ðñãø àú äåå÷èåøéí äðúåðéí áîèøéöä:
%\[A=(\a_1,\a_2,\a_3,\a_4)=
%\bm a_{11}&a_{12}&a_{13}&a_{14}\\
% 0&a_{22}&a_{23}&a_{24}\\
% 0&0&a_{33}&a_{34}\\
% 0&0&0&a_{44} \em.
%\]
%ëòú, 
%$\r(A)=4$
%ëé äî÷ãîéí 
%$a_{11}, a_{22}, a_{33}, a_{44}$ 
%ùåðéí îàôñ ìëï ëì àçã îåáéì åäîèøéöä îãåøâú. ìôé èòðä, ðåáò ù 
%$A\in \setR^{4\times 4}$
% äôéëä.
\end{enumerate}


	
\item 
éäé 
$(\setR^3, \setR)$ 
îøçá å÷èåøé òí äôòåìåú äøâéìåú åéäé 
$V=\{f\in \setR[x]\mid \deg f\leq 2\}$ 
îøçá äôåìéðåîéí îòì 
$\setR$ 
îîòìä ìëì äéåúø 
$2$.
ðâãéø ôåð÷öéä 
$T:\setR^3\to V$ 
òì éãé 	
$T\bm a\\b\\c \em=(a+c)x^2+(a-b)x+(b+c)$
ìëì 
$\bm a\\b\\c\em\in \setR^3$.

\begin{enumerate}
\item	
äåëéçå ù $T$ äòú÷ä ìéðàøéú.

%\answer
%
%éäéå
%$\u_1=\bm a_1\\b_1\\c_1\em, \u_2=\bm a_2\\b_2\\c_2\em\in \setR^3$
%åéäé 
%$\alpha\in \setR$. 
%àæé îú÷ééîéí 
%\begin{align*}T(\u_1+\u_2)&=T\bm a_1+a_2\\b_1+b_2\\c_1+c_2\em\\
%	&=(a_1+a_2+c_1+c_2)x^2+(a_1+a_2-b_1-b_2)x+(b_1+b_2+c_1+c_2)\\
%	&=(a_1+    c_1    )x^2+(a_1    -b_1    )x+b_1    +c_1+(a_2+    c_2    )x^2+(a_2    -b_2    )x+b_2    +c_2\\
%	&=T(\u_1)+T(\u_2)
%	\end{align*}
%	å 
%\begin{align*}T(\alpha \u_1)&=(\alpha a_1+   \alpha c_1    )x^2+(\alpha a_1    -\alpha b_1    )x+(\alpha b_1    +\alpha c_1    )\\
%	&=\alpha \Big(a_1+    c_1    )x^2+(a_1    -b_1    )x+b_1    +c_1\Big)\\
%	&=\alpha T(\u_1).
%\end{align*}
%ðåáò ëé 
%$T$ 
%äòú÷ä ìéðàøéú.

\item 
äøàå ëé 
$x^2+1\in \ip(T)$
åâí 
$-x+1\in \ip(T)$.

%\answer 
%
%ð÷ç 
%$a=b=1, c=0$ 
%åð÷áì 
%$T\bm 1\\1\\0 \em=(1+0)x^2+(1-1)x+1+0=x^2+1$ 
%ìëï 
%$x^2+1\in \ip(T)$.
%
%ð÷ç 
%$a=c=0, b=1$ 
%åð÷áì 
%$T\bm 0\\0\\1 \em=(0+0)x^2+(0-1)x+1+0=-x+1$ 
%ìëï 
%$-x+1\in \ip(T)$.


\item 
äåëéçå ëé 
$\dim (\ip(T))\geq 2$.

%\answer 
%
%áñòéó ä÷åãí øàéðå ëé 
%$x^2+1, -x+1\in \ip(T)$. 
%áøåø ùäí áú"ì ëé äîòìåú ùåðåú.
%îöã ùðé, 
%$\sp\{x^2+1, -x+1\}\subseteq \ip(T)$.
% ðåáò ëé 
%\[\dim (\ip(T))\geq \dim (\sp\{ x^2+1, -x+1 \})=2.\]

\item 
äøàå ëé 
$\bm 1\\1\\-1 \em\in \ker(T)$.


%\answer
%
%ðçùá 
%\[T\bm 1\\1\\-1 \em=(1-1)x^2+(1-1)x+1-1=0\]
%ìëï 
%$\bm 1\\1\\-1\em\in \ker T$.

\item 
äåëéçå ëé 
$\dim (\ker(T))\geq 1$.


%\answer
%
%ðåáò îäñòéó ä÷åãí ëé 
%$\sp\left\{ \bm 1\\1\\-1 \em \right\}\subseteq \ker T$
%ìëï 
%\[\dim(\ker T)\geq \dim \left(\sp\left\{ \bm 1\\1\\-1 \em \right\}\right)=1.\]

\item 
çùáå àú 
$\dim (\ker (T))$ 
åàú 
$\dim (\ip(T))$.


%\answer
%
%ìôé îùôè îú÷ééí 
%$\dim (\ker (T))+\dim (\ip(T))=\dim(\setR^3)=3$. 
%ìôé äñòéôéí ÷åãîéí 
%$\dim (\ker (T))\geq 1$ 
%å 
%$\dim (\ip(T))\geq 2$
%ìëï áäëøç 
%$\dim (\ker (T))= 1$ 
%å 
%$\dim (\ip(T))= 2$.

%\item 
%éäé 
%$B=\{x^2, x, 1\}$
%äáñéñ äñèðãøèé ùì 
%$V$.
%úäé  
%$A\in \setR^{3\times 3}$
%îèøéöä äî÷ééîú 
%$[T(\u)]_B=A\u$ 
%ìëì 
%$\u\in \setR^3$.
%\begin{enumerate}
%	\item 
%	îöàå àú 
%	$A$ 
%	áîôåøù.
% 
%%\answer
%%
%%ðëúåá 
%%$\u=\bm a\\b\\c \em$
%%åðçùá 
%%\[[T\u]_B=[(a+c)x^2+(a-b)x+(b+c)]_B=\bm a+c\\ a-b\\ b+c \em.\]
%%ðãøù ìîöåà îèøéöä 
%%$A$ 
%%ëê ù 
%%$A\u=A\bm a \\ b \\ c \em=\bm a+c\\ a-b\\ b+c \em$.
%%ðéúï ì÷çú 
%%\[A=\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\]
%%ëé àæ 
%%\[A\bm a \\ b \\ c \em=\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\bm a \\ b \\ c \em=\bm a+c\\ a-b\\ b+c \em.\]
%
%\item
%äøàå ëé 
%$\Nu(A)=\sp\left\{ \bm 1\\1\\-1 \em\right\}$.
%
%%\answer
%%
%%
%%ëòú, 
%%$\Nu(A)$ 
%%äåà àåñó äôúøåðåú ìîî"ì 
%%$A\x=\0$. 
%%ðôúåø: 
%%\begin{align*}
%%(A|\0)=\bse{ccc} 
%%1&0&1&0\\ 1&-1&0&0\\ 0&1&1&0 \ese &
%%\widesim{R_2\to R_2-R_1} 
%%\bse{ccc} 
%%1&0&1&0\\ 0&-1&-1&0\\ 0&1&1&0 \ese\\
%%&\widesim{R_3\to R_3+R_2} 
%%\bse{ccc} 
%%1&0&1&0\\ 0&-1&-1&0\\ 0&0&0&0 \ese
%%\widesim{R_2\to -R_2} 
%%\bse{ccc} 
%%1&0&1&0\\ 0&1&1&0\\ 0&0&0&0 \ese.
%%\end{align*}
%%
%%ðëúåá 
%%$c=t$ 
%%åàæ ð÷áì 
%%$a=b=-t$ 
%%ìëï éù ôúøåï ëììé 
%%$\bm a\\ b\\ c\em=\left\{ t\bm -1\\-1\\1 \em\mid t\in \setR \right\}$
%%ìëï 
%%\[\Nu(A)=\sp \left\{\bm -1\\-1\\1 \em\right\}=\sp \left\{\bm 1\\1\\-1 \em\right\}.\]
%
%\item 
%îöàå áñéñ ì 
%$\C(A)$.
%
%%\answer
%%
%%ìôé ääâãøä ùì îøçá òîåãåú
%%$\C(A)=\sp\left\{
%%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em, \bm 1\\0\\1 \em
%%\right\}.$
%%ëáø øàéðå ëé 
%%$\bm 1\\ 1\\ -1\em\in \Nu(A)$ 
%%ìëï 
%%\[\bm 0\\0\\0 \em=\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\bm 1\\ 1\\ -1\em=1\bm 1\\1\\0\em+1\bm 0\\ -1\\ 1 \em+(-1)\bm 1\\0\\1 \em.\]
%%ìëï ðåëì ìëúåá àú äòîåãä äàçøåðä ëöéøåó ìéðàøé ùì äòîåãåú äàçøåú: 
%%\[\bm 1\\0\\1 \em=\bm 1\\1\\0\em+\bm 0\\ -1\\ 1 \em.\]
%%ðåáò ëé 
%%$\C(A)=\sp\left\{
%%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em, \bm 1\\0\\1 \em
%%\right\}=\sp\left\{
%%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em
%%\right\}.$
%%äåå÷èåøéí 
%%$\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em$ 
%%äí áú"ì ëé äí àéðí î÷áéìéí ìëï äí îäååéí áñéñ ì 
%%$\C(A)$. 

%\end{enumerate}


\end{enumerate}


	\item 
	éäé 
	$\setF$ 
	ùãä. 
		\begin{enumerate}
		\item 
			äåëéçå ùàí 
			$A, B\in \setF^{n\times n}$ 
îèøéöåú ñéîèøéåú àæé
$AB$ 
ñéîèøéú àí åø÷ àí 
$AB=BA$.
		 

%\answer 
%
%ðúåï ù 
%$A, B$ 
%ñéîèøéåú ìëï 
%$A^T=A, B^T=B$. 
%ëòú, 
%$(AB)^T=B^TA^T=BA$. 
%ðéúï ìäñé÷ ëé 
%$AB$
%ñéîèøéú àí åø÷ àí 
%$(AB)^T=AB$ 
%àí åø÷ àí 
%$BA=AB$.		
		
		\item 
		äåëéçå ùàí 
		$A, B\in \setR^{3\times 3}$
		îèøéöåú ñéîèøéåú 
		ëê ù 
		$AB=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$
		àæé âí 
		$BA=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$.

%\answer
%
%äîèøéöä
%$AB=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$
%ñéîèøéú ëé äéà àìëñåðéú. ìôé äñòéó ä÷åãí ðåáò ëé 
%$BA=AB=\bm  2&0&0\\ 0&5&0\\ 0&0&-1 \em$.

\item 
úðå ãåâîä ìîèøéöåú 
$A, B\in \setR^{2\times 2}$
)ìà áäëøç ñéîèøéåú( ëê ù 
$AB$
ñéîèøéú å
${AB\neq BA}$.		

%\answer 
%
%ð÷ç 
%$A=\bm 1&1\\0&1 \em, B=\bm 1&0\\ 1&1 \em$.
%àæ
%$AB=\bm 1&1\\0&1 \em \bm 1&0\\ 1&1 \em=\bm 2&1\\1&1 \em$
%ñéîèøéú å 
%$BA=\bm 1&0\\ 1&1 \em \bm 1&1\\0&1 \em = \bm 1&1\\1&2 \em \neq AB$.

	\end{enumerate}

	\item \label{inverse-of-products}
	éäé 
	$\setF$
	ùãä åúäééðä 
	$A, B\in \setF^{n\times n}$
	îèøéöåú äôéëåú. 
	\begin{enumerate}
		\item 
		äåëéçå ù 
		$AB$ 
		äôéëä.

%\answer
%
%ìôé îùôè 
%$\det (AB)=\det A\det B$. 
%îëéååï ù 
%$A, B$ 
%äôéëåú îú÷ééí 
%$\det A, \det B\neq 0$. 
%ðåáò ëé 
%$\det (AB)\neq 0$ 
%ìëï 
%$AB$ 
%äôéëä.
		\item 
		äåëéçå ëé 
		$(AB)^{-1}=B^{-1}A^{-1}$.
		
%\answer		
%		
%òì îðú ìäøàåú ëé 
%$B^{-1}A^{-1}$ 
%ääåôëéú ùì $AB$ àðå ðçùá: 
%\[ (AB)(B^{-1}A^{-1})= A(BB^{-1})A^{-1}=AIA^{-1}=AA^{-1}=I. \]		
%ìôé îùôè, îú÷ééí âí 
%$(B^{-1}A^{-1})(AB)=I$
%ìëï 
%$B^{-1}A^{-1}$
%àëï ääåôëéú ùì 
%$AB$.

\item 

àí 
$C\in \setF^{n\times n}$
âí äôéëä, îöàå ðåñçä
òáåø 
$(ABC)^{-1}$
ùáä îåôéòéí äîèøéöåú 
$A^{-1}, B^{-1}, C^{-1}$.

%\answer 
%
%ìôé äñòéó ä÷åãí îú÷ééí 
%$(ABC)^{-1}=((AB)C)^{-1}=C^{-1}(AB)^{-1}=C^{-1}B^{-1}A^{-1}$.
%		
		\item 		
		äåëéçå ù 
		$A^2$ 
		äôéëä.
		
%\answer 
%
%ðåáò éùéøåú îñòéó à' ëàùø 
%$B=A$
%ëé àæ
%$AB=A^2$.		
		
		\item
		äåëéçå ù
		$(A^2)^{-1}=(A^{-1})^2$.
		
%\answer	
%		
%		
%		ð÷ç 
%		$B=A$ 
%		åàæ ìôé ñòéó ùì äùàìä äðåëçéú á' ð÷áì
%		\[(A^2)^{-1}=(AA)^{-1}=(A^{-1})(A^{-1})=(A^{-1})^2.\]

		\item 
		àí 
		\[A=\bm -11&3&-4\\ -18&5&-7\\ 8&-2&3 \em, B=\bm 1&-1&1\\1&2&5\\ -2&4&1 \em\]
		çùáå àú 
		$AB$ 
		åàú 
		$B^{-1}A^{-1}$.
	îåîìõ ìäéòæø áñòéó á' áî÷åí ìçùá àú 
		$A^{-1}$ 
	åàú 
		$B^{-1}$ 
		áðôøã.
		
%\answer		
%		
%\[AB=\bm 0&1&0\\1&0&0\\0&0&1 \em.\]		
%ìôé ñòéó á' îú÷ééí 
%$B^{-1}A^{-1}=(AB)^{-1}=\bm 0&1&0\\1&0&0\\0&0&1 \em^{-1}$.	
%ðçùá àú ääåôëéú:
%\begin{align*}
%	\left(
%	\begin{array}{ccc|ccc}
%		0&1&0&1&0&0\\
%		1&0&0&0&1&0\\
%		0&0&1&0&0&1
%	\end{array} 	
%	\right) 	
%	&{\widesim{R_1\leftrightarrow R_2}}	
%	\left(
%	\begin{array}{ccc|ccc}
%1&0&0&0&1&0\\
%0&1&0&1&0&0\\
%0&0&1&0&0&1
%	\end{array} 	
%	\right).
%\end{align*}
%	ðåáò ëé 
%	$B^{-1}A^{-1}=\bm 0&1&0\\1&0&0\\0&0&1 \em^{-1}=\bm 0&1&0\\1&0&0\\0&0&1 \em$.
	
	\end{enumerate}	
	
	
	\item \label{eigen-of-inverse}
	éäé 
	$\setF$
	ùãä åúäé 
	$A\in \setF^{n\times n}$
	îèøéöä äôéëä. 
\begin{enumerate}
\item 
äåëéçå ùëì òøê òöîé ùì 
$A$ 
ùåðä îàôñ.

%\answer
%
%ðúåï ù 
%$A$ 
%äôéëä. ìëï, ìôé èòðä, 
%$\Nu(A)=\{\0\}$. 
%ðåáò ù 
%$A\x\neq \0$ 
%ìëì 
%$\x\neq \0$.
%ìëï ìà éúëï ù 
%$A\x=0\x$ 
%òáåø å÷èåø 
%$\x\neq \0$.
%ðåáò ù 
%$0$ 
%àéðå òøê òöîé ùì 
%$A$.



\item 
äåëéçå ùàí 
$\x$ 
å÷èåø òöîé ùì 
$A$ 
òí òøê òöîé 
$\lambda$ 
àæé 
$\x$ 
å÷èåø òöîé ùì 
$A^{-1}$
òí òøê òöîé 
$\frac{1}{\lambda}$.
ëàï 
$A^{-1}$ 
îñîï àú äîèøéöä ääåôëéú ùì $A$.

%\answer
%
%ìôé äðúåï îú÷ééí 
%$A\x=\lambda \x$. 
%ðëôéì àú ùðé äàâôéí á 
%$A^{-1}$ 
%åð÷áì 
%\[A^{-1}A\x=A^{-1}(\lambda \x)=\lambda A^{-1}\x.\]
%ëòú, ìôé ääâãøä ùì îèøéöä äåôëéú îú÷ééí 
%$A^{-1}A=I$ 
%ëàùø 
%$I$ 
%îèøéöú äéçéãä îñãø 
%$n\times n$.
%ðåáò ù 
%$A^{-1}A\x=I\x=\x$ 
%ìëï 
%$\x=\lambda A^{-1}\x$.
%ðéúï ìäñé÷ ëé 
%$A^{-1}\x=\frac{1}{\lambda}\x$
%ìëï 
%$\x$ 
%å÷èåø òöîé ùì 
%$A^{-1}$ 
%òí òøê òöîé 
%$\frac{1}{\lambda}$.

\item 
äåëéçå ùàí 
$\x$ 
å÷èåø òöîé ùì 
$A$ 
òí òøê òöîé 
$\lambda$ 
àæé 
$\x$ 
å÷èåø òöîé ùì 
$A^2$
òí òøê òöîé 
                    $\lambda^2$.

%\answer
%
%ìôé äðúåï 
%$A\x=\lambda\x$.
%ëòú,
%$A^2\x=AA\x=A\lambda \x=\lambda A\x=\lambda (\lambda \x)=\lambda^2\x$
%ìëï 
%$\x$ 
%å÷èåø òöîé ùì 
%$A^2$ 
%òí òøê òöîé 
%$\lambda^2$.


\end{enumerate}

\item 

	éäé 
	$\setF$
	ùãä åúäé 
	$A\in \setF^{n\times n}$
	îèøéöä äôéëä. 

\begin{enumerate}
	\item 
äåëéçå ëé 
$(A^T)^{-1}=(A^{-1})^T$.	


%\answer
%
%ëãé ìáãå÷ äàí 
%$(A^{-1})^T$
%ääåôëéú ùì 
%$A^T$,
%ðçùá àú 
%$A^T(A^{-1})^T$. 
%ëòú, 
%\begin{align*}
%A^T(A^{-1})^T&=(A^{-1}A)^T\\
%&=I^T=I.
%\end{align*}
%ìôé îùôè ðåáò ùâí 
%$(A^{-1})^TA^T=I$.
%ìëï àëï 
%$(A^{-1})^T$
%ääåôëéú ùì 
%$A^T$.

	\item 
äåëéçå ùàí 
$A$ 
ñéîèøéú àæé âí 
$A^{-1}$
ñéîèøéú.

%\answer
%
%àí $A$ ñéîèøéú àæ 
%$A^T=A$. 
%ìôé ñòéó ä÷åãí ð÷áì 
%\[(A^{-1})^T=(A^T)^{-1}=A^{-1}\]
%ìëï 
%$A^{-1}$
%ñéîèøéú.
\item 
àí 
\[A=\bm 3&0&5\\0&2&1\\5&1&9 \em\]
\begin{enumerate}
	\item
çùáå àú 
$A^{-1}$.

%\answer
%
%òì îðú ìçùá àú ääåëôéú ùì $A$ ðãøâ:
%
%\begin{align*}
%	\left(
%	\begin{array}{ccc|ccc}
%		3&0&5&1&0&0\\
%		0&2&1&0&1&0\\
%		5&1&9&0&0&1
%	\end{array} 	
%	\right) 	
%	&{\widesim{R_3\to 3R_3-5R_1}}	
%	\left(
%	\begin{array}{ccc|ccc}
%		3&0&5&1&0&0\\
%		0&2&1&0&1&0\\
%		0&3&2&-5&0&3
%	\end{array} 	
%	\right)\\[0.5em]
%	&{\widesim{R_3\to 2R_3-3R_2}}	
%	\left(
%	\begin{array}{ccc|ccc}
%		3&0&5&1&0&0\\
%		0&2&1&0&1&0\\
%		0&0&1&-10&-3&6
%	\end{array} 	
%	\right)\\[0.5em]
%		&{\widesim{R_1\to R_1-5R_2}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			3&0&0&51&15&-30\\
%			0&2&1&0&1&0\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\[0.5em]
%		&{\widesim{R_2\to R_2-R_3}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			3&0&0&51&15&-30\\
%			0&2&0&10&4&-6\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\[0.5em]
%		&{\widesim{R_1\to \frac{1}{3}R_1}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			1&0&0&17&5&-10\\
%			0&2&0&10&4&-6\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\[0.5em]				
%		&{\widesim{R_2\to \frac{1}{2}R_2}}	
%		\left(
%		\begin{array}{ccc|ccc}
%			1&0&0&17&5&-10\\
%			0&1&0&5&2&-3\\
%			0&0&1&-10&-3&6
%		\end{array} 	
%		\right)\\						
%\end{align*}
%ìëï 
%$A^{-1}=\bm
%17&5&-10\\
%5&2&-3\\
%-10&-3&6
%\em$.

\item 
çùáå àú 
$(A^2)^{-1}$.

%\answer
%
%ìôé ùàìä 
%\ref{inverse-of-products}
% ñòéó ä' îú÷ééí 
%$(A^2)^{-1}=(A^{-1})^2$.
%ìôé äñòéó ä÷åãí ð÷áì 
%\begin{align*}
%(A^2)^{-1}&=(A^{-1})^2\\
%&=A^{-1}A^{-1}\\
%&=\bm
%17&5&-10\\
%5&2&-3\\
%-10&-3&6
%\em\bm
%17&5&-10\\
%5&2&-3\\
%-10&-3&6
%\em\\
%&=\bm
%414&125&-245\\
%125&38&-74\\
%-245&-74&145
%\em
%\end{align*}

\end{enumerate}
\end{enumerate}



\textbf{úæëåøú:}
òáåø îèøéöä äôéëä
$\bm p&q\\ r&s \em$
 îñãø 
$2\times 2$
äîèøéöä ääåôëéú ðúåðä òì éãé 
\[\bm p&q\\ r&s \em^{-1}=\frac{1}{ps-qr}\bm s&-q\\ -r&p \em.\]

\item 

éäéå
	$a, b\in \setR$ 
	ëê ùîú÷ééí 
	$a^2-b^2=1$.
	ðâãéø îèøéöåú 
	\[C=\bm a&b\\ b&a \em, D=\bm a&b\\ a&b \em \in \setR^{2\times 2}.\]

\begin{enumerate}
	\item 
	äøàå ù $D$ àéðä äôéëä.

%\answer 
%
%\textbf{ùéèä 1:}	
%$\det D=ab-ba=0$ 
%ìëï $D$ àéðä äôéëä.
%
%\textbf{ùéèä 2:}
%äòîåãåú ùì $D$ î÷áéìåú ìëï äï úìåéåú ìéðàøéú. ðåáò ëé 
%$D$ 
%àéðä äôéëä.
%
%
%\textbf{ùéèä 3:}
%äùåøåú ùì $D$ æäåú ìëï 
%$\r (D)<2$ 
%)áòöí 
%$\r (D)=1$ 
%ëé 
%$a^2-b^2=1$ 
%ìëï 
%$(a, b)\neq (0,0)$.(
%ðåáò ëé 
%$D$ 
%àéðä äôéëä.
%
%\textbf{ùéèä 4:}
%$D\bm b\\-a \em=\bm 0\\0 \em$
%ìëï 
%$\bm b\\ -a\em\in \Nu(D)$.
%àáì 
%$\bm b\\-a \em\neq \bm 0\\0 \em$
%ëé 
%$a^2-b^2=1$.
%ðåáò ëé 
%$\Nu(D)\neq \{\0\}$ 
%ìëï 
%$D$ 
%àéðä äôéëä.
	
	\item 
	äøàå ëé 
	$C$ 
	äôéëä åîöàå àú äîèøéöä ääåôëéú 
	$C^{-1}$.
	
%\answer	
%	
%ðçùá: 
%$\det C=a^2-b^2=1\neq 0$ 
%ìëï 
%$C$ 
%äôéëä. ìôé èòðä ääåôëéú ðúåðä òì éãé 
%$C^{-1}=\frac{1}{\det C}\bm a&-b\\ -b&a \em=\bm a&-b\\ -b&a \em$.
	
	\item 
	îöàå àú ëì äòøëéí äòöîééí ùì 
	$C$
)éù ìëúåá àú äò"ò áàîöòåú äôøîèøéí
$a, b$.(	
	òáåø ëì òøê òöîé îöàå å÷èåø òöîé îúàéí.
	
%\textbf{ùéèä 1:}
%ìîèøéöä 
%$C$ 
%éù äøáä ñéîèøéä. áôøè, ñëåí äî÷ãîéí áëì ùåøä ùååä. ðåáò ù 
%$\bm 1\\ 1\em$ 
%å÷èåø òöîé ùì 
%$C$ 
%àí òøê òöîé 
%$a+b$:
%
%\[C\bm 1\\ 1\em=\bm a&b\\ b&a \em\bm 1\\1 \em=\bm a+b\\ a+b \em=(a+b)\bm 1\\1 \em.\]	
%áàåôï ãåîä, äôøù äî÷ãîéí áëì ùåøä ãåîä ìëï ðáãå÷ àú äåå÷èåø 
%$\bm 1\\ -1\em$:
%
%\[C\bm 1\\ -1\em=\bm a&b\\ b&a \em\bm 1\\-1 \em=\bm a-b\\ b-a \em=(a-b)\bm 1\\-1 \em.\]
%ìîèøéöä îñãø 
%$2\times 2$ 
%éù ìëì äéåúø ùðé ùåøùéí ìôåìéðåí äàåôééðé ìëï éù ìëì äéåúø ùðé òøëéí òöîééí. ðéúï ìäñé÷ ùîöàðå àú ëì äòøëéí äòöîééí ùì $C$.
%\textbf{ùéèä 2:}
%ðçùá àú äôåìéðåí äàåôééðé ùì $C$:
%\begin{align*}
%p_C(\lambda)&=\bd a-\lambda&b\\ b&a-\lambda \ed\\
%&=(a-\lambda)^2-b^2\\
%&=\lambda^2-2a\lambda+a^2-b^2\\
%&=(\lambda-(a+b))(\lambda-(a-b))	
%\end{align*}	
%ìëï éù òøëéí òöîééí 
%$a+b,\ \ a-b$.
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$a+b$ 
%ðîöà å÷èåø ìà àôñ á 
%$\Nu(A-(a+b)I)$:
%\begin{align*}
%(A-(a+b)I|\0)&=\bse{cc} -b&b&0\\ b&-b&0 \ese{\widesim{R_2\to R_2+R_1}}\bse{cc} -b&b&0\\ 0&0&0 \ese
%\end{align*}	
%àæ ðéúï ì÷çú å"ò
%$\v_1=\bm1\\1 \em$.\\
%
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$a-b$ 
%ðîöà å÷èåø ìà àôñ á 
%$\Nu(A-(a-b)I)$:
%\begin{align*}
%	(A-(a-b)I|\0)&=\bse{cc} b&b&0\\ b&b&0 \ese{\widesim{R_2\to R_2-R_1}}\bse{cc} b&b&0\\ 0&0&0 \ese
%\end{align*}	
%àæ ðéúï ì÷çú å"ò
%$\v_2=\bm1\\-1 \em$.
		\item 
		îöàå àú ëì äòøëéí äòöîééí ùì 
		$D$
		)éù ìëúåá àú äò"ò áàîöòåú äôøîèøéí
		$a, b$.(
		òáåø ëì òøê òöîé îöàå å÷èåø òöîé îúàéí.
		
%\textbf{ùéèä 1:}
%ëîå áñòéó ä÷åãí, ìîèøéöä 
%$D$ 
%éù äøáä ñéîèøéä. áôøè, ñëåí äî÷ãîéí áëì ùåøä ùååä. ðåáò ù 
%$\bm 1\\ 1\em$ 
%å÷èåø òöîé ùì 
%$D$ 
%àí òøê òöîé 
%$a+b$:
%
%\[D\bm 1\\ 1\em=\bm a&b\\ a&b \em\bm 1\\1 \em=\bm a+b\\ a+b \em=(a+b)\bm 1\\1 \em.\]	
%ëòú, ðéúï ìàôñ àú ùúé äùåøåú àí ðëôéì áåå÷èåø
%$\bm b\\ -a\em$
%ìëï äåà å÷èåø òöîé òí òøê òöîé 
%$0$:
%
%\[D\bm b\\ -a\em=\bm a&b\\ a&b \em\bm b\\-a \em=\bm ab-ba\\ ab-ba \em=\bm 0\\0 \em=0\bm b\\-a \em.\]
%ìîèøéöä îñãø 
%$2\times 2$ 
%éù ìëì äéåúø ùðé ùåøùéí ìôåìéðåí äàåôééðé ìëï éù ìëì äéåúø ùðé òøëéí òöîééí. ðéúï ìäñé÷ ùîöàðå àú ëì äòøëéí äòöîééí ùì $D$.
%\textbf{ùéèä 2:}
%ðçùá àú äôåìéðåí äàåôééðé ùì $D$:
%\begin{align*}
%	p_D(\lambda)&=\bd a-\lambda&b\\ a&b-\lambda \ed\\
%	&=(a-\lambda)(b-\lambda)-ab\\
%	&=\lambda^2-(a+b)\lambda\\
%	&=\lambda(\lambda-(a+b))
%\end{align*}	
%ìëï éù òøëéí òöîééí 
%$a+b,\ \ 0$.
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$a+b$ 
%ðîöà å÷èåø ìà àôñ á 
%$\Nu(A-(a+b)I)$:
%\begin{align*}
%	(A-(a+b)I|\0)&=\bse{cc} -b&b&0\\ a&-a&0 \ese{\widesim{}}\bse{cc} 1&-1&0\\ 0&0&0 \ese
%\end{align*}	
%îëéååï ù 
%$a^2-b^2=1$ 
%ìëï 
%$a\neq 0$ 
%àå 
%$b\neq 0$ 
%åäùåøåú î÷áéìåú.
%àæ ðéúï ì÷çú å"ò
%$\v_1=\bm1\\1 \em$.\\
%
%òì îðú ìîöåà å÷èåø òöîé òáåø òøê äòöîé 
%$0$ 
%ðîöà å÷èåø ìà àôñ á 
%${\Nu(A-(0)I)}$:
%\begin{align*}
%	(A-(0)I|\0)=(A|\0)&=\bse{cc} a&b&0\\ a&b&0 \ese{\widesim{R_2\to R_2-R_1}}\bse{cc} a&b&0\\ 0&0&0 \ese
%\end{align*}	
%àæ ðéúï ì÷çú å"ò
%$\v_2=\bm b\\-a \em$
%åäåà àéðå å÷èåø äàôñ ëé 
%$a\neq 0$ 
%àå 
%$b\neq 0$ 
%ëôé ùëáø öééðå.		
\item 
	äøàå ëé éù ì 
	$C$ 
	å 
	$C^{-1}$ 
	àåúí òøëéí òöîééí.
	
%	\answer
%	
%	ìôé ñòéó á' ùì äùàìä äðåëçéú äîèøéöä 
%	$C$ 
%	äôéëä.
%	ìôé ùàìä 
%	\ref{eigen-of-inverse}
%	ñòéó á' ìëì òøê òöîé 
%	$\lambda$ 
%	ùì $C$ éù ì
%	$C^{-1}$ 
%	éùðí òøëéí òöîééí 
%	 $\frac{1}{\lambda}$.
%	 ìôé ñòéó â' ùì äùàìä äðåëçéú äòøëéí äòöîééí ùì $C$ äí 
%	 $a+b,\ a-b$. 
%	 ðåáò ùäòøëéí äòöîééí ùì 
%	 $C^{-1}$ 
%	 äí 
%	 $\frac{1}{a+b}, \ \frac{1}{a-b}$.
%	 ðúåï ëé 
%	 $1=a^2-b^2=(a+b)(a-b)$
%	 ìëï
%	 $\frac{1}{a+b}=a-b$
%	 å 
%	 $\frac{1}{a-b}=a+b$.
%	 ìëï ì 
%	 $C^{-1}$ 
%éù àåúí òøëéí òöîééí 
%	 $a+b, \ a-b$ 
%	 ëîå 
%	 $C$.
%	
	\item 
	úðå ãåâîä ìîèøéöä äôéëä $Y$ îñãø 
	$2\times 2$ 
	ëê ùäòøëéí òöîééí ùì 
	$Y$ 
	ùåðéí îäòøëéí äòöîééí ùì 
	$Y^{-1}$.
	
	
%	\answer
%	
%ð÷ç ìîùì
%$Y=\bm 2&0\\ 0&3 \em$
%åàæ 
%$Y^{-1}=\bm \frac{1}{2}&0\\ 0&\frac{1}{3} \em$.
%äòøëéí äòöîééí ùì 
%$Y$ 
%äí 
%$2, 3$
%)ðéúï ì÷çú ìîùì å÷èåøéí òöîééí 
%$\bm 1\\ 0\em, \bm 0\\ 1\em$.(
%áàåôï ãåîä, ÷ì ìøàåú ùäòøëéí äòöîééí ùì 
%$Y^{-1}$ 
%äí 
%$\frac{1}{2}, \frac{1}{3}$. 
%ìëï ì 
%$Y, Y^{-1}$
%éù òøëéí òöîééí ùåðéí.
	
\end{enumerate}

\item 
ðâãéø ÷áåöú îèøéöåú òì éãé 
\[\S=\left\{ \bm a&-b\\ b&a  \em\mid a,b\in \setR, a^2+b^2=1 \right\}.\]
\begin{enumerate}
	\item 
	äøàå ùëì îèøéöä á 
	$\S$ 
	äôéëä.
	
%	\answer 
%	
%àí 
%$A=\bm a&-b\\ b&a\em\in \S$ 
%àæ 
%$\det A=a^2-(-b^2)=a^2+b^2=1\neq 0$ 
%ìëï 
%$A$ 
%äôéëä.

	\item 
	äåëéçå ùìëì 
	$A\in \S$ 
	îú÷ééí 
	$A^{-1}\in \S$.
	
%\answer	
%	
%àí 
%$A=\bm a&-b\\ b&a\em\in \S$ 	
%	àæ ìôé èòðä îú÷ééí 
%\[A^{-1}=\frac{1}{\det A}\bm a&b\\ -b&a \em=\frac{1}{a^2+b^2}\bm a&b\\ -b&a \em=\frac{1}{1}\bm a&b\\ -b&a \em=\bm a&-(-b)\\ b&a \em\in \S\]
%)äúðàé 
%$a^2+b^2=1$
%òãééï îú÷ééí.(	
	
	\item 
	äåëéçå ùìëì 
	$A, B\in \S$
	îú÷ééí 
	$AB\in \S$.
	
%\answer
%
%úééðä 
%	$A=
%	\bm a&-b\\ b&a \em, B=
%	\bm c&-d\\ d&c \em.
%	$
%	
%	
%	ëòú, 
%	\begin{align*}
%	AB&=\bm a&-b\\ b&a \em
%	\bm c&-d\\ d&c \em=\bm ac-bd&-ad-bc \\bc+ad&ac-bd  \em	
%	\end{align*}
	
	\item 
	îöàå ëì äôúøåðåú ìîùååàä 
	$A^2=\bm 1&0\\ 0&1 \em$
	ëàùø 
	$A\in \S$.
	
%\answer 
%
%ð÷ç 
%$a=-1, b=0$ 
%åð÷áì 	
%	$A=\bm a&-b\\b&a \em=\bm -1&0\\0&-1 \em$.
%	îëàï ð÷áì 
%	\[A^2=\bm -1&0\\0&-1 \em^2=\bm -1&0\\0&-1 \em\bm -1&0\\0&-1 \em=\bm 1&0\\0&1 \em\]
%	
\end{enumerate}

%\item 
%
%\begin{enumerate}
%	\item 
%	\item 
%\end{enumerate}




%\item 
%
%\begin{enumerate}
%	\item 
%	\item 
%\end{enumerate}





\end{enumerate}



\textbf{äåøàåú:}

\begin{itemize}
	\item
îèìä æå àéðä ìäâùä.
	\item 
ôúøåðåú éôåøñîå ìëì äîàåçø á 11 áéåìé.

\end{itemize}













\end{document}

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Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, Eli Zaretskii <eliz@gnu.org>,
> 	emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 15:17:12 +0300
> 
>    > Try renaming the file to document.bin and resend it as octet-stream...
> 
> Thanks for pointing it out! I will do that 

This file is encoded in codepage 1255, the MS-Windows Hebrew
encoding.  So you can display it correctly like this:

  C-x RET c cp1255 RET C-x C-f document RET

You could then save it in UTF-8:

  C-x RET f utf-8 RET
  C-x C-s

Alternatively, you could recode externally using 'iconv', but using
"-f CP1255 -t UTF-8".

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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>>> "EZ" == Eli Zaretskii <eliz@gnu.org> writes:

   >> From: Uwe Brauer <oub@mat.ucm.es>
   >> Cc: Uwe Brauer <oub@mat.ucm.es>, Eli Zaretskii <eliz@gnu.org>,
   >> emacs-devel@gnu.org
   >> Date: Thu, 18 Jul 2019 15:17:12 +0300
   >> 
   >> > Try renaming the file to document.bin and resend it as octet-stream...
   >> 
   >> Thanks for pointing it out! I will do that 

   > This file is encoded in codepage 1255, the MS-Windows Hebrew
   > encoding.  So you can display it correctly like this:

   >   C-x RET c cp1255 RET C-x C-f document RET

   > You could then save it in UTF-8:

   >   C-x RET f utf-8 RET
   >   C-x C-s

Thanks I will try this out.


   > Alternatively, you could recode externally using 'iconv', but using
   > "-f CP1255 -t UTF-8".

But why does 
iconv -f ISO8859-8 -t UTF-8 also work?

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Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, larsi@gnus.org,
> 	emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 16:03:54 +0300
> 
> But why does 
> iconv -f ISO8859-8 -t UTF-8 also work?

Because, as Andreas points out, the characters in the file have the
same encoding in both ISO8859-8 and CP1255.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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>>> "EZ" == Eli Zaretskii <eliz@gnu.org> writes:

   >> From: Uwe Brauer <oub@mat.ucm.es>
   >> Cc: Uwe Brauer <oub@mat.ucm.es>, Eli Zaretskii <eliz@gnu.org>,
   >> emacs-devel@gnu.org
   >> Date: Thu, 18 Jul 2019 15:17:12 +0300
   >> 
   >> > Try renaming the file to document.bin and resend it as octet-stream...
   >> 
   >> Thanks for pointing it out! I will do that 

   > This file is encoded in codepage 1255, the MS-Windows Hebrew
   > encoding.  So you can display it correctly like this:

   >   C-x RET c cp1255 RET C-x C-f document RET


That works, thanks, BUT


C-x RET c iso8859-8 RET C-x C-f document RET

Also works!

Two comments

    1. Are these  codings  basically the same?

    2. Why can't emacs detect and display these files automatically?
       Besides if I did not know the coding in advance what were I
       supposed to do?


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Re: problem with old hebrew latex files in iso 8859-8 coding

[Andreas Schwab]

On Jul 18 2019, Uwe Brauer <oub@mat.ucm.es> wrote:

>     2. Why can't emacs detect and display these files automatically?

It is impossible to distinguish single-byte encodings.

>        Besides if I did not know the coding in advance what were I
>        supposed to do?

Guess.

Andreas.

-- 
Andreas Schwab, SUSE Labs, schwab@suse.de
GPG Key fingerprint = 0196 BAD8 1CE9 1970 F4BE  1748 E4D4 88E3 0EEA B9D7
"And now for something completely different."

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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>>> "AS" == Andreas Schwab <schwab@suse.de> writes:

   > On Jul 18 2019, Uwe Brauer <oub@mat.ucm.es> wrote:
   >> 2. Why can't emacs detect and display these files automatically?

   > It is impossible to distinguish single-byte encodings.

Hm in the old days of pre mule, I am pretty sure emacs opened these
files correctly.


   >> Besides if I did not know the coding in advance what were I
   >> supposed to do?

   > Guess.

Ok, but I still don't understand why 

iconv -f ISO8859-8 -t UTF-8 

AND


iconv -f CP1255 -t UTF-8 

Work

 is CP1255 and iso8859-8 the SAME coding?

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Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

   > Hm in the old days of pre mule, I am pretty sure emacs opened these
   > files correctly.




   > Ok, but I still don't understand why 

   > iconv -f ISO8859-8 -t UTF-8 

   > AND


   > iconv -f CP1255 -t UTF-8 

   > Work

   >  is CP1255 and iso8859-8 the SAME coding?

To answer my question myself:

https://en.wikipedia.org/wiki/Windows-1255

Ok I learned something. 

The only remaining bit  of doubts concerns my memory. Did pre mule emacs
opened and displayed these files correctly.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Date: Thu, 18 Jul 2019 16:52:17 +0300
> Cc: Uwe Brauer <oub@mat.ucm.es>, Eli Zaretskii <eliz@gnu.org>, larsi@gnus.org,
>  emacs-devel@gnu.org
> 
> Hm in the old days of pre mule, I am pretty sure emacs opened these
> files correctly.

In pre-MULE days Emacs didn't decode the file's text at all, it just
displayed them with your default font.  The pre-MULE Emacs could only
display a single encoding and a single language/script.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, larsi@gnus.org,
> 	emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 16:14:15 +0300
> 
> C-x RET c iso8859-8 RET C-x C-f document RET
> 
> Also works!
> 
> Two comments
> 
>     1. Are these  codings  basically the same?

They are very close, yes.

>     2. Why can't emacs detect and display these files automatically?

It's impossible to distinguish between single-byte encodings without
elaborate linguistic analysis, which need sufficiently large text (and
Emacs doesn't know how to do that anyway).

>        Besides if I did not know the coding in advance what were I
>        supposed to do?

You are supposed to guess based on what you know about the files.

[Warning and iso 8859-1 default?] (was: problem with old hebrew latex files in iso 8859-8 coding)

[Uwe Brauer]

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> They are very close, yes.


> It's impossible to distinguish between single-byte encodings without
> elaborate linguistic analysis, which need sufficiently large text (and
> Emacs doesn't know how to do that anyway).

I just run the following experiment (the result you might respect, but I
found it surprising).

    1. I open a new file.

    2. I write שלום איל

    3. I save it as iso8859-8 (no warning pops up)

    4. I reopen the file and its interpreted as iso8859-1

Firstly, why does Emacs expect this to be iso8859-1? Is there any variable
I could change?

Secondly, shouldn't Emacs send a warning: «you are saving this file as
single-byte, be aware that when emacs reopen the file it might not
guess the coding» or something like this???

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Re: [Warning and iso 8859-1 default?] (was: problem with old hebrew latex files in iso 8859-8 coding)

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Date: Sat, 20 Jul 2019 16:39:19 +0300
> 
>     1. I open a new file.
> 
>     2. I write שלום איל
> 
>     3. I save it as iso8859-8 (no warning pops up)
> 
>     4. I reopen the file and its interpreted as iso8859-1
> 
> Firstly, why does Emacs expect this to be iso8859-1? Is there any variable
> I could change?

It's because of your locale, most probably.

> Secondly, shouldn't Emacs send a warning: «you are saving this file as
> single-byte, be aware that when emacs reopen the file it might not
> guess the coding» or something like this???

That'd be annoying, as these encodings are still used quite widely.

Re: [Warning and iso 8859-1 default?]

[Uwe Brauer]

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> It's because of your locale, most probably.
Hm I have 
setenv LC_ALL en_US.UTF8 

So I am still puzzled

> That'd be annoying, as these encodings are still used quite widely.

ok

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Re: [Warning and iso 8859-1 default?]

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Date: Sat, 20 Jul 2019 16:57:41 +0300
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> 
> > It's because of your locale, most probably.
> Hm I have 
> setenv LC_ALL en_US.UTF8 
> 
> So I am still puzzled

Why are you puzzled?  Nothing in your locale says to prefer 8859-8.

What does coding-system-priority-list return?

Re: [Warning and iso 8859-1 default?]

[Uwe Brauer]

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   > Why are you puzzled?  Nothing in your locale says to prefer 8859-8.

Yes but it does not say anything abiout 8859-1 either 

   > What does coding-system-priority-list return?

(utf-8 iso-2022-7bit iso-latin-1 iso-2022-7bit-lock iso-2022-8bit-ss2 emacs-mule raw-text iso-2022-jp in-is13194-devanagari chinese-iso-8bit utf-8-auto utf-8-with-signature ...)

So iso-latin-1 is the third entry, which explains the behaviour. 
3 questions then

    1. Where is this configured so that I can change it?

    2. What is emacs-mule

    3. What is utf-8-with-signature

Thanks 

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Re: [Warning and iso 8859-1 default?]

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> Date: Sat, 20 Jul 2019 17:16:37 +0300
> 
>    > Why are you puzzled?  Nothing in your locale says to prefer 8859-8.
> 
> Yes but it does not say anything abiout 8859-1 either 

Latin-1 is a reasonable general-purpose default, whereas ISO-8859-8
isn't.

>    > What does coding-system-priority-list return?
> 
> (utf-8 iso-2022-7bit iso-latin-1 iso-2022-7bit-lock iso-2022-8bit-ss2 emacs-mule raw-text iso-2022-jp in-is13194-devanagari chinese-iso-8bit utf-8-auto utf-8-with-signature ...)
> 
> So iso-latin-1 is the third entry, which explains the behaviour. 

Yes.

> 3 questions then
> 
>     1. Where is this configured so that I can change it?

set-coding-system-priority (but you really should learn to read the
manual, so you could answer these questions yourself).

>     2. What is emacs-mule
> 
>     3. What is utf-8-with-signature

"C-h C-\" will describe them (as it will any other coding-system).

Re: [Warning and iso 8859-1 default?]

[Uwe Brauer]

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   > Latin-1 is a reasonable general-purpose default, whereas ISO-8859-8
   > isn't.

Well for me most likely yes, for the friend of mine, who is reading
mostly english and hebrew, may 8859-8 might be  reasonable.

   > Yes.


   > set-coding-system-priority (but you really should learn to read the
   > manual, so you could answer these questions yourself).

This is what I thought but it is not an interactive function. I thought
I could do it via customize, but I cannot.
This is the documentation
,----
| 
| set-coding-system-priority is a built-in function in ‘C source code’.
| 
| (set-coding-system-priority &rest coding-systems)
| 
| Assign higher priority to the coding systems given as arguments.
| If multiple coding systems belong to the same category,
| all but the first one are ignored.
`----

And this 
(set-coding-system-priority (utf-8 iso-2022-7bit iso-latin-1 iso-2022-7bit-lock iso-2022-8bit-ss2 emacs-mule raw-text iso-2022-jp in-is13194-devanagari chinese-iso-8bit utf-8-auto utf-8-with-signature))

(set-coding-system-priority '(utf-8 iso-2022-7bit iso-latin-1
iso-2022-7bit-lock iso-2022-8bit-ss2 emacs-mule raw-text iso-2022-jp
in-is13194-devanagari chinese-iso-8bit utf-8-auto utf-8-with-signature))

Does not work. Sorry but the documentation is not helpful, at least not
for me.

Also not 
https://www.gnu.org/software/emacs/manual/html_node/elisp/Specifying-Coding-Systems.html




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Re: [Warning and iso 8859-1 default?]

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> Date: Sat, 20 Jul 2019 19:17:36 +0300
> 
>    > set-coding-system-priority (but you really should learn to read the
>    > manual, so you could answer these questions yourself).
> 
> This is what I thought but it is not an interactive function. I thought
> I could do it via customize, but I cannot.
> This is the documentation
> ,----
> | 
> | set-coding-system-priority is a built-in function in ‘C source code’.
> | 
> | (set-coding-system-priority &rest coding-systems)
> | 
> | Assign higher priority to the coding systems given as arguments.
> | If multiple coding systems belong to the same category,
> | all but the first one are ignored.
> `----
> 
> And this 
> (set-coding-system-priority (utf-8 iso-2022-7bit iso-latin-1 iso-2022-7bit-lock iso-2022-8bit-ss2 emacs-mule raw-text iso-2022-jp in-is13194-devanagari chinese-iso-8bit utf-8-auto utf-8-with-signature))
> 
> (set-coding-system-priority '(utf-8 iso-2022-7bit iso-latin-1
> iso-2022-7bit-lock iso-2022-8bit-ss2 emacs-mule raw-text iso-2022-jp
> in-is13194-devanagari chinese-iso-8bit utf-8-auto utf-8-with-signature))

The docs says

  (set-coding-system-priority &rest coding-systems)

  Assign higher priority to the coding systems given as arguments.
                                ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
So you should call the function with coding-systems, not with a list
of coding-systems:

  (set-coding-system-priority 'utf-8 'iso-2022-7bit 'iso-latin-1 'iso-2022-7bit-lock 'iso-2022-8bit-ss2 'emacs-mule 'raw-text 'iso-2022-jp 'in-is13194-devanagari 'chinese-iso-8bit 'utf-8-auto 'utf-8-with-signature)

And be sure to read this part:

  If multiple coding systems belong to the same category,
  all but the first one are ignored.

Re: [Warning and iso 8859-1 default?]

[Eli Zaretskii]

> Date: Sat, 20 Jul 2019 19:46:02 +0300
> From: Eli Zaretskii <eliz@gnu.org>
> Cc: emacs-devel@gnu.org
> 
>   (set-coding-system-priority 'utf-8 'iso-2022-7bit 'iso-latin-1 'iso-2022-7bit-lock 'iso-2022-8bit-ss2 'emacs-mule 'raw-text 'iso-2022-jp 'in-is13194-devanagari 'chinese-iso-8bit 'utf-8-auto 'utf-8-with-signature)

Alternatively, you could use prefer-coding-system, which is an
interactive command.

Re: [Warning and iso 8859-1 default?]

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Date: Sat, 20 Jul 2019 19:17:36 +0300
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> 
>    > Latin-1 is a reasonable general-purpose default, whereas ISO-8859-8
>    > isn't.
> 
> Well for me most likely yes, for the friend of mine, who is reading
> mostly english and hebrew, may 8859-8 might be  reasonable.

How should Emacs know that?  How would it know that your setup is
different from his?  The locale's codeset is the only hint Emacs gets
from the environment, everything else is left to user customizations.

Re: [Warning and iso 8859-1 default?]

[Stefan Monnier]

>    > Latin-1 is a reasonable general-purpose default, whereas ISO-8859-8
>    > isn't.
> Well for me most likely yes, for the friend of mine, who is reading
> mostly english and hebrew, may 8859-8 might be  reasonable.

And that's why utf-8 is now the recommended coding-system.


        Stefan

Re: [Warning and iso 8859-1 default?]

[Uwe Brauer]

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   > And that's why utf-8 is now the recommended coding-system.

Right, but for a person who has hundreds of iso-8859-8 files

(set-coding-system-priority 'utf-8 'iso-2022-7bit 'iso-8859-8 'iso-2022-7bit-lock 'iso-2022-8bit-ss2 'emacs-mule 'raw-text 'iso-2022-jp 'in-is13194-devanagari 'chinese-iso-8bit 'utf-8-auto 'utf-8-with-signature)


Might be better than



(set-coding-system-priority 'utf-8 'iso-2022-7bit 'iso-latin-1 'iso-2022-7bit-lock 'iso-2022-8bit-ss2 'emacs-mule 'raw-text 'iso-2022-jp 'in-is13194-devanagari 'chinese-iso-8bit 'utf-8-auto 'utf-8-with-signature)




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Re: [Warning and iso 8859-1 default?]

[Stefan Monnier]

>     1. I open a new file.
>     2. I write שלום איל
>     3. I save it as iso8859-8 (no warning pops up)

How did you do this step 3?


        Stefan

Re: [Warning and iso 8859-1 default?]

[Uwe Brauer]

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>>> "SM" == Stefan Monnier <monnier@iro.umontreal.ca> writes:

   >> 1. I open a new file.
   >> 2. I write שלום איל
   >> 3. I save it as iso8859-8 (no warning pops up)

   > How did you do this step 3?

(set-buffer-file-coding-system 'iso-8859-8 nil)
save-buffer 
kill-buffer

Uwe 

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Re: [Warning and iso 8859-1 default?]

[Stefan Monnier]

> (set-buffer-file-coding-system 'iso-8859-8 nil)
> save-buffer

I see, then I guess the only place where we could detect the problem is
in `save-buffer`.  Not sure how we could do it cheaply enough to be
worth the trouble.


        Stefan

Re: [Warning and iso 8859-1 default?]

[Eli Zaretskii]

> From: Stefan Monnier <monnier@iro.umontreal.ca>
> Date: Sat, 20 Jul 2019 13:27:58 -0400
> 
> > (set-buffer-file-coding-system 'iso-8859-8 nil)
> > save-buffer
> 
> I see, then I guess the only place where we could detect the problem is
> in `save-buffer`.

What problem is that?  He saved Hebrew text with encoding that is
suitable for Hebrew text.

Re: [Warning and iso 8859-1 default?]

[Stefan Monnier]

>> I see, then I guess the only place where we could detect the problem is
>> in `save-buffer`.
> What problem is that?

The problem of saving the file in an encoding that will be mis-detected
when read back using that same Emacs.


        Stefan

Re: [Warning and iso 8859-1 default?]

[Eli Zaretskii]

> From: Stefan Monnier <monnier@iro.umontreal.ca>
> Cc: emacs-devel@gnu.org
> Date: Sat, 20 Jul 2019 13:37:04 -0400
> 
> >> I see, then I guess the only place where we could detect the problem is
> >> in `save-buffer`.
> > What problem is that?
> 
> The problem of saving the file in an encoding that will be mis-detected
> when read back using that same Emacs.

That would make saving using any single-byte encoding "a problem",
right?

Rather than calling this a problem we should perhaps work on improving
detection of single-byte encodings.

Re: [Warning and iso 8859-1 default?]

[Stefan Monnier]

>> >> I see, then I guess the only place where we could detect the problem is
>> >> in `save-buffer`.
>> > What problem is that?
>> The problem of saving the file in an encoding that will be mis-detected
>> when read back using that same Emacs.
> That would make saving using any single-byte encoding "a problem",
> right?

Not quite: there's usually one single-byte encoding that's correctly
detected (iso-8859-1 in Uwe's case).

BTW, when I saw "a problem" I don't mean something we should prevent,
but Uwe expected to be warned about it somewhere along the way.

> Rather than calling this a problem we should perhaps work on improving
> detection of single-byte encodings.

Single-byte encodings are pretty rare nowadays, so I'm not sure it's
worth the trouble.  But maybe you're right (e.g. maybe the spread of
utf-8 makes it less frequent for people to have the expertise needed to
guess the right encoding when it's not utf-8).


        Stefan

Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 15:06:43 +0300
> 
> > I don't understand what files you attached -- are those the files
> > presumably encoded in ISO8859-8, or after conversion to UTF-8?  What I
> > see are UTF-8 encoded files, but the characters in them are not Hebrew
> > characters, they are Latin characters.
> 
> These are the original files, for me GNU emacs present them as they were
> latin-1 encoded, I attach a screen shot.
> Please note the 1 in the left lower corner.

I see the same display after telling Emacs to decode the file as
UTF-8.  Also, the command 'file' agrees with me:

  document.txt: UTF-8 Unicode text, with CRLF line terminators

In any case, it should be clear that displaying Latin character for
what's supposed to be Hebrew text is not what you'd expect, right?

Please send me the file which, when visited in Emacs, shows the
display of the image you posted.  It cannot be the file you posted
originally, because that one is clearly in UTF-8.

> > Can you send the original files, before any conversion/mangling?
> 
> These are the original files.

Please re-verify that.  Maybe it's your mailer that did some recoding,
in which case please send the files as a compressed archive.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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   > Please re-verify that.  Maybe it's your mailer that did some recoding,
   > in which case please send the files as a compressed archive.

As Lars pointed out, gnus converted them to UTF8.

I just resent them as recommended
  > Try renaming the file to document.bin and resend it as octet-stream...


As I said I convert the original files with 


iconv -f ISO8859-8 -t UTF-8 old.txt > new.txt

And the new one emacs displays perfectly. So I conclude the old ones
were in iso8859-8 encoding.



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Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 15:19:46 +0300
> 
> As I said I convert the original files with 
> 
> 
> iconv -f ISO8859-8 -t UTF-8 old.txt > new.txt
> 
> And the new one emacs displays perfectly. So I conclude the old ones
> were in iso8859-8 encoding.

No, they were in codepage-1255.  See my other message.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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>>> "EZ" == Eli Zaretskii <eliz@gnu.org> writes:

>> 
>> And the new one emacs displays perfectly. So I conclude the old ones
>> were in iso8859-8 encoding.

> No, they were in codepage-1255.  See my other message.

Hm that is confusing. Please try the following

Open a new file and write

שלום אלי

Save it as utf8, say test.txt

Now run 

iconv -f UTF-8 -t ISO8859-8  testu8.txt > test8859.txt

Open test8859.txt in emacs and you face the same problem.

So what is the conclusion? Iconv does not work correctly?


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Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 15:55:47 +0300
> 
> > No, they were in codepage-1255.  See my other message.
> 
> Hm that is confusing. Please try the following
> 
> Open a new file and write
> 
> שלום אלי
> 
> Save it as utf8, say test.txt
> 
> Now run 
> 
> iconv -f UTF-8 -t ISO8859-8  testu8.txt > test8859.txt
> 
> Open test8859.txt in emacs and you face the same problem.

I don't see any problem when I do that.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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   > I don't see any problem when I do that.

That is very odd. Attached two screenshots (and I used emacs -q to be
sure it was something in my .emacs file which caused the misbehaviour)

I don't see the converted file correctly in emacs. You do? What is your
setting?


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Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 17:04:46 +0300
> 
>    > I don't see any problem when I do that.
> 
> That is very odd. Attached two screenshots (and I used emacs -q to be
> sure it was something in my .emacs file which caused the misbehaviour)
> 
> I don't see the converted file correctly in emacs. You do? What is your
> setting?

How (with what sequence of commands) did you visit the file?  In your
screenshots it is clear that Emacs thought this file was encoded in
ISO-8859-1, which is why you see Latin characters.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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   > How (with what sequence of commands) did you visit the file?  In your
   > screenshots it is clear that Emacs thought this file was encoded in
   > ISO-8859-1, which is why you see Latin characters.

Very innocently with C-x C-f

I did not use (on purpose your recommendation of)
  C-x RET c cp1255 RET C-x C-f document RET

But maybe you did this, and then we have a misunderstanding.

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Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 17:26:10 +0300
> 
>    > How (with what sequence of commands) did you visit the file?  In your
>    > screenshots it is clear that Emacs thought this file was encoded in
>    > ISO-8859-1, which is why you see Latin characters.
> 
> Very innocently with C-x C-f

Then you cannot expect anything else: Emacs basically uses your
locale's default 8-bit encoding when you do that.

> I did not use (on purpose your recommendation of)
>   C-x RET c cp1255 RET C-x C-f document RET
> 
> But maybe you did this

Of course I did.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Andreas Schwab]

On Jul 18 2019, Eli Zaretskii <eliz@gnu.org> wrote:

>> And the new one emacs displays perfectly. So I conclude the old ones
>> were in iso8859-8 encoding.
>
> No, they were in codepage-1255.  See my other message.

How do you know?  They are identical for the characters in the files.

Andreas.

-- 
Andreas Schwab, SUSE Labs, schwab@suse.de
GPG Key fingerprint = 0196 BAD8 1CE9 1970 F4BE  1748 E4D4 88E3 0EEA B9D7
"And now for something completely different."

Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Andreas Schwab <schwab@suse.de>
> Cc: Uwe Brauer <oub@mat.ucm.es>,  emacs-devel@gnu.org
> Date: Thu, 18 Jul 2019 15:05:08 +0200
> 
> On Jul 18 2019, Eli Zaretskii <eliz@gnu.org> wrote:
> 
> >> And the new one emacs displays perfectly. So I conclude the old ones
> >> were in iso8859-8 encoding.
> >
> > No, they were in codepage-1255.  See my other message.
> 
> How do you know?  They are identical for the characters in the files.

Ah, you are right.  The gibberish I saw with ISO-8859-8 was with the
files recoded into UTF-8 by Gnus.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Andreas Schwab]

On Jul 18 2019, Uwe Brauer <oub@mat.ucm.es> wrote:

> The problem is that while his files are displayed perfectly in texstudio
> (having iso 8859-8 enabled), which means the system has the requires
> fonts installed, emacs fails to do so. It simply does not display the
> hebrew chars with the correct fonts.

Since single-byte encodings cannot be detected automatically, you have
to tell Emacs explicitly how the file is encoded.

> What does work is to open the file in GNU emacs, save it as UTF-8, the
> resulting file is still not correctly displayed, I even tried to save
> the file as 8859-8 but then I received the following error
>
>
>
> ,----
> | These default coding systems were tried to encode text
> | in the buffer ‘doc2.txt’:
> |   (hebrew-iso-8bit-unix (1 . 233) (2 . 228) (3 . 233) (7 . 238) (8 .
> |   248) (9 . 231) (10 . 225) (12 . 229) (14 . 232) ...)
> | However, each of them encountered characters it couldn’t encode:
> |   hebrew-iso-8bit-unix cannot encode these: é ä é î ø ç á å è å ...

That means Emacs has used latin-1 as the default single-byte encoding.

Andreas.

-- 
Andreas Schwab, SUSE Labs, schwab@suse.de
GPG Key fingerprint = 0196 BAD8 1CE9 1970 F4BE  1748 E4D4 88E3 0EEA B9D7
"And now for something completely different."

Re: problem with old hebrew latex files in iso 8859-8 coding

[Stefan Monnier]

> A friend of mine posses a lot of hebrew latex files which still use
> 8859-8 coding. Understandable he wants to switch to UTF8 and xelatex.
[...]
> \documentclass[11pt]{article}
> %\usepackage[8859-8]{inputenc}

How does his TeX system know that this file is using 8859-8?
Normally, the `inputenc` package above does that, but it's
commented out.

If you uncomment it, Emacs should also automatically recognize that the
file is using iso-8859-8 and display the file as intended.


        Stefan

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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>>> "SM" == Stefan Monnier <monnier@iro.umontreal.ca> writes:

>> A friend of mine posses a lot of hebrew latex files which still use
>> 8859-8 coding. Understandable he wants to switch to UTF8 and xelatex.
> [...]
>> \documentclass[11pt]{article}
>> %\usepackage[8859-8]{inputenc}

> How does his TeX system know that this file is using 8859-8?
> Normally, the `inputenc` package above does that, but it's
> commented out.


I asked this myself, and came to the conclusion 
the culmus package does this. But I am not sure

> If you uncomment it, Emacs should also automatically recognize that the
> file is using iso-8859-8 and display the file as intended.

Good point tried it out. Using texstudio to save it.

But still Emacs does not recognize this file as hebrew.

Uwe 

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Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> From: Uwe Brauer <oub@mat.ucm.es>
> Date: Thu, 18 Jul 2019 18:24:55 +0300
> Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> 
> > If you uncomment it, Emacs should also automatically recognize that the
> > file is using iso-8859-8 and display the file as intended.
> 
> Good point tried it out. Using texstudio to save it.
> 
> But still Emacs does not recognize this file as hebrew.

Because latex-inputenc-coding-alist doesn't have a match for 8859-8.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Eli Zaretskii]

> Date: Thu, 18 Jul 2019 19:06:30 +0300
> From: Eli Zaretskii <eliz@gnu.org>
> Cc: oub@mat.ucm.es, monnier@iro.umontreal.ca, emacs-devel@gnu.org
> 
> > From: Uwe Brauer <oub@mat.ucm.es>
> > Date: Thu, 18 Jul 2019 18:24:55 +0300
> > Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
> > 
> > > If you uncomment it, Emacs should also automatically recognize that the
> > > file is using iso-8859-8 and display the file as intended.
> > 
> > Good point tried it out. Using texstudio to save it.
> > 
> > But still Emacs does not recognize this file as hebrew.
> 
> Because latex-inputenc-coding-alist doesn't have a match for 8859-8.

And for a good reason: according to
http://mirrors.ctan.org/macros/latex/base/inputenc.pdf, there's no
such encoding as 8859-8, the inputenc package doesn't support it.

Re: problem with old hebrew latex files in iso 8859-8 coding

[Arne Jørgensen]

Eli Zaretskii <eliz@gnu.org> writes:

>> Date: Thu, 18 Jul 2019 19:06:30 +0300
>> From: Eli Zaretskii <eliz@gnu.org>
>> Cc: oub@mat.ucm.es, monnier@iro.umontreal.ca, emacs-devel@gnu.org
>> 
>> > From: Uwe Brauer <oub@mat.ucm.es>
>> > Date: Thu, 18 Jul 2019 18:24:55 +0300
>> > Cc: Uwe Brauer <oub@mat.ucm.es>, emacs-devel@gnu.org
>> > 
>> > > If you uncomment it, Emacs should also automatically recognize that the
>> > > file is using iso-8859-8 and display the file as intended.
>> > 
>> > Good point tried it out. Using texstudio to save it.
>> > 
>> > But still Emacs does not recognize this file as hebrew.
>> 
>> Because latex-inputenc-coding-alist doesn't have a match for 8859-8.
>
> And for a good reason: according to
> http://mirrors.ctan.org/macros/latex/base/inputenc.pdf, there's no
> such encoding as 8859-8, the inputenc package doesn't support it.

It might have existed at some point:
https://ctan.math.illinois.edu/macros/latex/contrib/oberdiek/inputenx.pdf
(dated 2016/05/16). My old LaTeX installation also has it.

You could try:

  (add-to-list 'latex-inputenc-coding-alist '("8859-8" . iso-8859-8))


/Arne

Re: problem with old hebrew latex files in iso 8859-8 coding

[Uwe Brauer]

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   > Eli Zaretskii <eliz@gnu.org> writes:

   > It might have existed at some point:
   > https://ctan.math.illinois.edu/macros/latex/contrib/oberdiek/inputenx.pdf
   > (dated 2016/05/16). My old LaTeX installation also has it.

   > You could try:

   >   (add-to-list 'latex-inputenc-coding-alist '("8859-8" . iso-8859-8))

Interesting, thanks for pointing it out!

Uwe 

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