From mboxrd@z Thu Jan 1 00:00:00 1970 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Uwe Brauer Newsgroups: gmane.emacs.devel Subject: problem with old hebrew latex files in iso 8859-8 coding Date: Thu, 18 Jul 2019 14:31:44 +0300 Message-ID: <87zhlbtw7z.fsf@mat.ucm.es> Mime-Version: 1.0 Content-Type: multipart/mixed; boundary="=-=-=" Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="175392"; mail-complaints-to="usenet@blaine.gmane.org" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/27.0.50 (gnu/linux) To: emacs-devel@gnu.org Original-X-From: emacs-devel-bounces+ged-emacs-devel=m.gmane.org@gnu.org Thu Jul 18 13:32:16 2019 Return-path: Envelope-to: ged-emacs-devel@m.gmane.org Original-Received: from lists.gnu.org ([209.51.188.17]) by blaine.gmane.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.89) (envelope-from ) id 1ho4e0-000jUe-54 for ged-emacs-devel@m.gmane.org; Thu, 18 Jul 2019 13:32:16 +0200 Original-Received: from localhost ([::1]:36624 helo=lists1p.gnu.org) by lists.gnu.org with esmtp (Exim 4.86_2) (envelope-from ) id 1ho4dy-0005ks-SY for ged-emacs-devel@m.gmane.org; Thu, 18 Jul 2019 07:32:14 -0400 Original-Received: from eggs.gnu.org ([2001:470:142:3::10]:51822) by lists.gnu.org with esmtp (Exim 4.86_2) (envelope-from ) id 1ho4dg-0005jz-Ug for emacs-devel@gnu.org; Thu, 18 Jul 2019 07:32:00 -0400 Original-Received: from Debian-exim by eggs.gnu.org with spam-scanned (Exim 4.71) (envelope-from ) id 1ho4dd-0001j6-6r for emacs-devel@gnu.org; Thu, 18 Jul 2019 07:31:56 -0400 Original-Received: from mail-wm1-x32a.google.com ([2a00:1450:4864:20::32a]:51659) by eggs.gnu.org with esmtps (TLS1.0:RSA_AES_128_CBC_SHA1:16) (Exim 4.71) (envelope-from ) id 1ho4dc-0001g6-D7 for emacs-devel@gnu.org; Thu, 18 Jul 2019 07:31:53 -0400 Original-Received: by mail-wm1-x32a.google.com with SMTP id 207so25242348wma.1 for ; Thu, 18 Jul 2019 04:31:51 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=mat.ucm.es; s=google; h=from:to:subject:date:message-id:user-agent:mime-version; bh=BXAeSQA0NvL3BhzfTkClqyCsTWTqzyTJ7z8mrlBadC8=; b=XGT//DMT+AQeCtEtSKB/HHhPqkJc7+dClcTm8udzz84IE+mQMHMiHpn3VSA8oTkMPj bXcA11/swR4tqddf5vsMmkVYOB03gVNhAWF+kKKnxOLmwmgg/UwVKNV8uq/5jxJ/KnQN eZAMk77diVlMv2sTPGECShI9HCpzTuNVnaS4g= X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20161025; h=x-gm-message-state:from:to:subject:date:message-id:user-agent :mime-version; bh=BXAeSQA0NvL3BhzfTkClqyCsTWTqzyTJ7z8mrlBadC8=; b=rhBRs45bHNtA8vqy+JG/E0+4TdTdDug1sCGXWBWzGlrvFCNrB4c98BtVx6aVNk5IlM /aO96fDjO/EZA9s1bC1AdTgoaems5xla+H+0W0qdSpCKgleQnqcMMOD8R+OOYqwQQRnJ N+RvadhTvgAPygEnmudse7ZHSxoxFsZhsu9NCvetV69TIv78hHUs7xxjLpWH++Np5hIB udmz2qlj1Ksxx16xdCLSdnQMeqTLYG6/f7zwi1tWUgRuKZt/WgBBJbaC3Rwp7yrrCucr WvjOtlHe0gh24UDFzD/zUqDhTqkCJGngxY5LLWulfOrwkIy0SNXklr73v6eWwVk3TkAs 8ILA== X-Gm-Message-State: APjAAAVUxntTG13EGyIYNyK9UZkS/XR7K8g7sKiZxuUnq3ntAnNpiQn5 t2LNvAHR2RU1lsWgMemohglFZ7/970tj2g== X-Google-Smtp-Source: APXvYqz4YF1YHF4PRTSA+fFckPJ9FQwozWp18fhc/2+8wjo6lOPnkUaFVSJRr0CN4xaySpJtRuaPBw== X-Received: by 2002:a1c:f505:: with SMTP id t5mr11207260wmh.67.1563449509070; Thu, 18 Jul 2019 04:31:49 -0700 (PDT) Original-Received: from Utnapischtim (vpn-219-248.vpn.ucm.es. [147.96.219.248]) by smtp.gmail.com with ESMTPSA id t6sm26685131wmb.29.2019.07.18.04.31.47 for (version=TLS1_2 cipher=ECDHE-RSA-AES128-GCM-SHA256 bits=128/128); Thu, 18 Jul 2019 04:31:48 -0700 (PDT) X-Mailer: emacs 27.0.50 (via feedmail 11-beta-1 I) X-detected-operating-system: by eggs.gnu.org: Genre and OS details not recognized. X-Received-From: 2a00:1450:4864:20::32a X-BeenThere: emacs-devel@gnu.org X-Mailman-Version: 2.1.23 Precedence: list List-Id: "Emacs development discussions." List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Errors-To: emacs-devel-bounces+ged-emacs-devel=m.gmane.org@gnu.org Original-Sender: "Emacs-devel" Xref: news.gmane.org gmane.emacs.devel:238635 Archived-At: --=-=-= Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable Hello=20 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=20 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 =E2=80=98doc2.txt=E2=80=99: | (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=E2=80=99t encode: | hebrew-iso-8bit-unix cannot encode these: =C3=A9 =C3=A4 =C3=A9 =C3=AE = =C3=B8 =C3=A7 =C3=A1 =C3=A5 =C3=A8 =C3=A5 ... |=20 | Click on a character (or switch to this window by =E2=80=98C-x o=E2=80=99 | and select the characters by RET) to jump to the place it appears, | where =E2=80=98C-u C-x =3D=E2=80=99 will give information about it. |=20 | 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). |=20 | 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=20 --=-=-= Content-Type: text/plain; charset=utf-8 Content-Disposition: attachment; filename=document.txt Content-Transfer-Encoding: base64 w6nDpMOpIA0KDQrDrsO4w6fDoSDDpcO3w6jDpcO4w6kgw6vDoMO5w7ggDQoNCsOlIA0KDQrDocOx w6nDsSDDrCAgDQrDpMOlw6vDqcOnw6Ugw6DDpSDDpMO0w7jDqcOrw6Ugw6DDuiDDpMOow7LDsMOl w7ogw6TDocOgw6XDui4gDQoNCg0Kw6zDq8OsIMOlw7fDqMOlw7ggDQoNCsOkw7fDocOlw7bDpCAN Cg0Kw6HDscOpw7Egw6wgDQo= --=-=-= Content-Type: text/x-tex; charset=utf-8 Content-Disposition: attachment; filename=junk.tex Content-Transfer-Encoding: quoted-printable \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 =C3=B0=C3=B7'(}} \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} =C3=B0=C3=B7'(} \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 {=C3=A1=C3=A9=C3=A7=C3=B1 =C3=AC=C3=B4=C3=B2=C3=A5=C3=AC=C3= =A5=C3=BA =C3=A4=C3=B8=C3=A2=C3=A9=C3=AC=C3=A5=C3=BA} \def \pageoflines {\notes[24pt]{18}{\textwidth}} \def \answer {\textbf{=C3=B4=C3=BA=C3=B8=C3=A5=C3=AF:}} \def \S {\mathcal{S}} %\newtheorem{theorem}{Theorem} \newtheorem{theorem}{\R{=C3=AE=C3=B9=C3=B4=C3=A8}} \newtheorem{example}{\R{=C3=A3=C3=A5=C3=A2=C3=AE=C3=A4}} %\newtheorem{question}{\R{=C3=B9=C3=A0=C3=AC=C3=A4}} \theorembodyfont{\fontfamily{david}\selectfont}\newtheorem{question}{\R{ \fontfamily{david}\selectfont \textbf{=C3=B9=C3=A0=C3=AC=C3=A4}}} \theorembodyfont{\fontfamily{david}\selectfont}\newtheorem{solution}{\R{ \fontfamily{david}\selectfont \textbf{=C3=B4=C3=BA=C3=B8=C3=A5=C3=AF}}} %\newtheorem{remark}[theorem]{Remark} %\newcommand{{ =C3=B9=C3=A0=C3=AC=C3=A4}}{\noindent{\bf \R{ .=C3=B9=C3=A0= =C3=AC=C3=A4}}} %------------------------------------------------------------------------ \begin{document} %\fontfamily{david}\selectfont \begin{center} {\textbf{ \Large{ =C3=AE=C3=A8=C3=AC=C3=A4 =C3=AE=C3=B1' 5 =C3=A0=C3=AC=C3=A2=C3=A1=C3=B8=C3= =A4 1=C3=AE=C3=A7' \L{11102} \\ \quad \\ =C3=AC=C3=A0 =C3=AC=C3=A4=C3=A2=C3=B9=C3=A4 }} \\} \end{center} % %\textbf{=C3=A4=C3=B2=C3=B8=C3=A4:} %=C3=A1=C3=AE=C3=A8=C3=AC=C3=A4 =C3=A6=C3=A5 =C3=AE=C3=A5=C3=BA=C3=B8 =C3= =AC=C3=A4=C3=B9=C3=BA=C3=AE=C3=B9 =C3=A1=C3=AB=C3=AC =C3=A8=C3=B2=C3=B0=C3= =A4 =C3=B9=C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=A1=C3=AE=C3=A8=C3=AC=C3= =A5=C3=BA =C3=B7=C3=A5=C3=A3=C3=AE=C3=A5=C3=BA. \begin{enumerate} \item =C3=A9=C3=A4=C3=A9=20 $(V, \setF)$ =C3=AE=C3=B8=C3=A7=C3=A1 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=C3=A9 =C3=AB=C3=A0= =C3=B9=C3=B8=20 $\dim V=3Dn$ =C3=A5=20 $B=3D\{\v_1, \ldots, \v_n\}$=20 =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC $V$. =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=A0=C3=A5 =C3=A4=C3=B4=C3=B8=C3=A9= =C3=AB=C3=A5 =C3=A0=C3=BA =C3=A4=C3=A8=C3=B2=C3=B0=C3=A5=C3=BA =C3=A4=C3=A1= =C3=A0=C3=A5=C3=BA.=20 \begin{enumerate} \item=20 =C3=AC=C3=AB=C3=AC =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=20 $\y\in V$=20 =C3=A4=C3=B7=C3=A1=C3=A5=C3=B6=C3=A4=20 $\{\y+\v_1, \ldots, \y+\v_n\}$=20 =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC $V$. %\answer=20 % %=C3=A4=C3=A8=C3=B2=C3=B0=C3=A4 =C3=A0=C3=A9=C3=B0=C3=A4 =C3=B0=C3=AB=C3=A5= =C3=B0=C3=A4. =C3=B0=C3=B7=C3=A7=20 %$\y=3D-\v_1$=20 %=C3=A5=C3=A0=C3=A6 =C3=A4=C3=B7=C3=A1=C3=A5=C3=B6=C3=A4=20 %\[\{\y+\v_1, \ldots, \y+\v_n\}=3D\{-\v_1+\v_1, \ldots, -\v_1+\v_n\}=3D\{\0= , \ldots, -\v_1+\v_n\}\] %=C3=AE=C3=AB=C3=A9=C3=AC=C3=A4 =C3=A0=C3=BA =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8= =C3=A4=C3=A0=C3=B4=C3=B1 =C3=AC=C3=AB=C3=AF =C3=A4=C3=A9=C3=A0 =C3=A0=C3= =A9=C3=B0=C3=A4 =C3=A1=C3=BA"=C3=AC )=C3=AB=C3=AE=C3=A5=C3=A1=C3=AF =C3=A4= =C3=A9=C3=A0 =C3=A2=C3=AD =C3=AC=C3=A0 =C3=A1=C3=B1=C3=A9=C3=B1(. \item=20 =C3=AC=C3=AB=C3=AC =C3=B1=C3=B7=C3=AC=C3=B8=C3=A9=C3=AD=20 $\alpha_1, \ldots, \alpha_n\neq 0$ =C3=A4=C3=B7=C3=A1=C3=A5=C3=B6=C3=A4 $\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$=09 =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC=20 $V$. %\answer % %=C3=A4=C3=B7=C3=A1=C3=A5=C3=B6=C3=A4 %$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$=09 %=C3=AE=C3=BA=C3=B7=C3=A1=C3=AC=C3=BA =C3=AE=C3=A4=C3=A1=C3=B1=C3=A9=C3=B1 = $B$ =C3=B2=C3=AC =C3=A9=C3=A3=C3=A9 =C3=B1=C3=A3=C3=B8=C3=A4 =C3=B9=C3=AC = =C3=B4=C3=B2=C3=A5=C3=AC=C3=A5=C3=BA =C3=A0=C3=AC=C3=AE=C3=B0=C3=A8=C3=B8= =C3=A9=C3=A5=C3=BA. =C3=AC=C3=B4=C3=A9 =C3=A8=C3=B2=C3=B0=C3=A4, =C3=B4=C3= =B2=C3=A5=C3=AC=C3=A5=C3=BA =C3=A0=C3=AC=C3=AE=C3=B0=C3=A8=C3=B8=C3=A9=C3= =A5=C3=BA =C3=AC=C3=A0 =C3=AE=C3=B9=C3=B0=C3=A5=C3=BA =C3=BA=C3=AC=C3=A5=C3= =BA =C3=AC=C3=A9=C3=B0=C3=A0=C3=B8=C3=A9=C3=BA =C3=A0=C3=A5 =C3=B4=C3=B8=C3= =A9=C3=B9=C3=A4 =C3=B9=C3=AC =C3=B7=C3=A1=C3=A5=C3=B6=C3=A4. =C3=AB=C3=AC= =C3=AF %$\{\alpha_1 \v_1, \ldots, \alpha_n \v_n\}$ % =C3=A2=C3=AD =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC $V$. \item=20 =C3=A1=C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=A6=C3=A4 =C3=A1=C3=AC=C3=A1=C3=A3 $V=3D\setR^n$ =C3=AE=C3=B8=C3=A7=C3=A1 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=C3=A9 =C3=AE=C3=B2= =C3=AC =C3=A4=C3=B9=C3=A3=C3=A4=20 $\setR$=20 \usualop\=20 =C3=A5=20 ${B=3D\{\v_1, \ldots, \v_n\}}$ =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC=20 $\setR^n$. =C3=B2=C3=A1=C3=A5=C3=B8 $A\in \setR^{n\times n}$=20 =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=AE=C3=B1=C3=A3=C3=B8=20 $n\times n$ =C3=A4=C3=B7=C3=A1=C3=A5=C3=B6=C3=A4=20 $\{A\v_1, \ldots, A\v_n\}$=20 =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC $V$ =C3=A0=C3=AD =C3=A5=C3=B8=C3=B7 =C3=A0= =C3=AD=20 $A$=20 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. %\answer % %=C3=B0=C3=A6=C3=AB=C3=A9=C3=B8 =C3=AB=C3=A9=20 %$A$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A4 =C3=A0=C3=AD =C3=A5=C3=B8=C3=B7 =C3=A0=C3= =AD=20 %$\Nu(A)=3D\{\0\}$. % %\textbf{=C3=AB=C3=A9=C3=A5=C3=A5=C3=AF 1:} %=C3=B0=C3=B0=C3=A9=C3=A7 =C3=AB=C3=A9 $A$ =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. %=C3=A9=C3=A4=C3=A9=C3=A5=20 %$\alpha_1, \ldots, \alpha_n\in \setR $ %=C3=AB=C3=AA =C3=B9 %\begin{align*} %\alpha_1 A\v_1+\cdots +\alpha_n A\v_n=3D\0. %\end{align*} %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %\[A(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=3D\0.\] %=C3=AE=C3=AB=C3=A9=C3=A5=C3=A5=C3=AF =C3=B9=20 %$A$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A4 =C3=B0=C3=A5=C3=AB=C3=AC =C3=AC=C3=A4=C3=AB= =C3=B4=C3=A9=C3=AC =C3=A1=C3=B9=C3=B0=C3=A9 =C3=A4=C3=A0=C3=A2=C3=B4=C3=A9= =C3=AD =C3=A1=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=A5=C3=B4=C3=AB= =C3=A9=C3=BA=20 %$A^{-1}$ %=C3=A5=C3=B0=C3=B7=C3=A1=C3=AC %\[A^{-1}A(\alpha_1 \v_1+\cdots +\alpha_n \v_n)=3DI(\alpha_1 \v_1+\cdots +\= alpha_n \v_n)=3D\alpha_1 \v_1+\cdots +\alpha_n \v_n=3D\0\] %=C3=A0=C3=A1=C3=AC $B$ =C3=A1=C3=BA"=C3=AC )=C3=AB=C3=A9 =C3=A4=C3=A5=C3= =A0 =C3=A1=C3=B1=C3=A9=C3=B1( =C3=AC=C3=AB=C3=AF %$\alpha_1=3D\cdots=3D\alpha_n=3D0$. %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %$A\v_1,\ldots, A\v_n$ %=C3=A1=C3=BA"=C3=AC =C3=A0=C3=A1=C3=AC=20 %$\dim V=3Dn$=20 %=C3=AC=C3=AB=C3=AF=20 %$A\v_1,\ldots, A\v_n$ %=C3=AE=C3=A4=C3=A5=C3=A5=C3=A9=C3=AD =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC=20 %$V$. % %\textbf{=C3=AB=C3=A9=C3=A5=C3=A5=C3=AF 2:} % %=C3=B0=C3=B0=C3=A9=C3=A7 =C3=B9=20 %$A\v_1,\ldots, A\v_n$ %=C3=AE=C3=A4=C3=A5=C3=A5=C3=A9=C3=AD =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC=20 %$\setR^n$. %=C3=B2=C3=AC =C3=AE=C3=B0=C3=BA =C3=AC=C3=A4=C3=B8=C3=A0=C3=A5=C3=BA =C3= =B9 $A$ =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4, =C3=B0=C3=B8=C3=A0=C3=A4 =C3=B9 %$\Nu(A)=3D\{\0\}$. %=C3=AC=C3=B9=C3=AD =C3=AB=C3=AA, =C3=A9=C3=A4=C3=A9=20 %$\x\in \Nu(A)$. %=C3=AE=C3=AB=C3=A9=C3=A5=C3=A5=C3=AF=20 %$B$ %=C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC=20 %$\setR^n$,=20 %=C3=A4=C3=A9=C3=A0 =C3=B4=C3=A5=C3=B8=C3=B9=C3=BA =C3=A0=C3=BA=20 %$\setR^n$. %=C3=AC=C3=AB=C3=AF =C3=B7=C3=A9=C3=A9=C3=AE=C3=A9=C3=AD=20 %$\beta_1, \ldots, \beta_n\in \setR$=20 %=C3=AB=C3=AA =C3=B9=20 %$\beta_1\v_1+\cdots+\beta_n \v_n=3D\x$. %=C3=B0=C3=AB=C3=B4=C3=A9=C3=AC =C3=A1=20 %$A$=20 %=C3=A5=C3=B0=C3=B7=C3=A1=C3=AC=20 %\begin{align*} %&A(\beta_1\v_1+\cdots+\beta_n \v_n)=3DA\x=3D\0\\ %\implies & \beta_1A\v_1+\cdots+\beta_n A\v_n=3D\0. %\end{align*} %=C3=AE=C3=AB=C3=A9=C3=A5=C3=A5=C3=AF =C3=B9=20 %$\{A\v_1,\ldots, A\v_n\}$ %=C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC=20 %$\setR^n$=20 %=C3=A4=C3=A9=C3=A0 =C3=A1=C3=BA"=C3=AC. =C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3= =A9=20 %$\beta_1=3D\cdots=3D\beta_n=3D0$.=20 %=C3=AE=C3=BA=C3=B7=C3=A1=C3=AC =C3=B9=20 %$\x=3D\beta_1\v_1+\cdots+\beta_n\v_n=3D\0$. %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %$\Nu(A)=3D\{\0\}$=20 %=C3=AC=C3=AB=C3=AF=20 %$A$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. \item=20 =C3=A9=C3=A4=C3=A9=C3=A5=20 $\a_1=3D\bm a_{11}\\ 0\\0\\0 \em, \a_2=3D\bm a_{12}\\ a_{22}\\0\\0 \em, \a_= 3=3D\bm a_{13}\\ a_{23}\\a_{33}\\0 \em, \a_4=3D\bm a_{14}\\ a_{24}\\a_{34}\= \a_{44} \em$ =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=C3=A9=C3=AD =C3=A1=20 $\setR^4$=20 =C3=AB=C3=A0=C3=B9=C3=B8=20 $a_{11}, a_{22}, a_{33}, a_{44}\neq 0$.=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=A4=C3=A5=C3=A5=C3=B7=C3=A8= =C3=A5=C3=B8=C3=A9=C3=AD =C3=AE=C3=A4=C3=A5=C3=A5=C3=A9=C3=AD =C3=A1=C3=B1= =C3=A9=C3=B1 =C3=AC=20 $\setR^4$. %\answer % %=C3=B0=C3=B1=C3=A3=C3=B8 =C3=A0=C3=BA =C3=A4=C3=A5=C3=A5=C3=B7=C3=A8=C3=A5= =C3=B8=C3=A9=C3=AD =C3=A4=C3=B0=C3=BA=C3=A5=C3=B0=C3=A9=C3=AD =C3=A1=C3=AE= =C3=A8=C3=B8=C3=A9=C3=B6=C3=A4: %\[A=3D(\a_1,\a_2,\a_3,\a_4)=3D %\bm a_{11}&a_{12}&a_{13}&a_{14}\\ % 0&a_{22}&a_{23}&a_{24}\\ % 0&0&a_{33}&a_{34}\\ % 0&0&0&a_{44} \em. %\] %=C3=AB=C3=B2=C3=BA,=20 %$\r(A)=3D4$ %=C3=AB=C3=A9 =C3=A4=C3=AE=C3=B7=C3=A3=C3=AE=C3=A9=C3=AD=20 %$a_{11}, a_{22}, a_{33}, a_{44}$=20 %=C3=B9=C3=A5=C3=B0=C3=A9=C3=AD =C3=AE=C3=A0=C3=B4=C3=B1 =C3=AC=C3=AB=C3=AF= =C3=AB=C3=AC =C3=A0=C3=A7=C3=A3 =C3=AE=C3=A5=C3=A1=C3=A9=C3=AC =C3=A5=C3= =A4=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=AE=C3=A3=C3=A5=C3=B8=C3=A2=C3= =BA. =C3=AC=C3=B4=C3=A9 =C3=A8=C3=B2=C3=B0=C3=A4, =C3=B0=C3=A5=C3=A1=C3=B2 = =C3=B9=20 %$A\in \setR^{4\times 4}$ % =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. \end{enumerate} =09 \item=20 =C3=A9=C3=A4=C3=A9=20 $(\setR^3, \setR)$=20 =C3=AE=C3=B8=C3=A7=C3=A1 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=C3=A9 =C3=B2=C3=AD = =C3=A4=C3=B4=C3=B2=C3=A5=C3=AC=C3=A5=C3=BA =C3=A4=C3=B8=C3=A2=C3=A9=C3=AC= =C3=A5=C3=BA =C3=A5=C3=A9=C3=A4=C3=A9=20 $V=3D\{f\in \setR[x]\mid \deg f\leq 2\}$=20 =C3=AE=C3=B8=C3=A7=C3=A1 =C3=A4=C3=B4=C3=A5=C3=AC=C3=A9=C3=B0=C3=A5=C3=AE= =C3=A9=C3=AD =C3=AE=C3=B2=C3=AC=20 $\setR$=20 =C3=AE=C3=AE=C3=B2=C3=AC=C3=A4 =C3=AC=C3=AB=C3=AC =C3=A4=C3=A9=C3=A5=C3=BA= =C3=B8=20 $2$. =C3=B0=C3=A2=C3=A3=C3=A9=C3=B8 =C3=B4=C3=A5=C3=B0=C3=B7=C3=B6=C3=A9=C3=A4=20 $T:\setR^3\to V$=20 =C3=B2=C3=AC =C3=A9=C3=A3=C3=A9=20=09 $T\bm a\\b\\c \em=3D(a+c)x^2+(a-b)x+(b+c)$ =C3=AC=C3=AB=C3=AC=20 $\bm a\\b\\c\em\in \setR^3$. \begin{enumerate} \item=09 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9 $T$ =C3=A4=C3=B2=C3=BA=C3=B7=C3= =A4 =C3=AC=C3=A9=C3=B0=C3=A0=C3=B8=C3=A9=C3=BA. %\answer % %=C3=A9=C3=A4=C3=A9=C3=A5 %$\u_1=3D\bm a_1\\b_1\\c_1\em, \u_2=3D\bm a_2\\b_2\\c_2\em\in \setR^3$ %=C3=A5=C3=A9=C3=A4=C3=A9=20 %$\alpha\in \setR$.=20 %=C3=A0=C3=A6=C3=A9 =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AE=C3=A9=C3=AD=20 %\begin{align*}T(\u_1+\u_2)&=3DT\bm a_1+a_2\\b_1+b_2\\c_1+c_2\em\\ % &=3D(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)\\ % &=3D(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\\ % &=3DT(\u_1)+T(\u_2) % \end{align*} % =C3=A5=20 %\begin{align*}T(\alpha \u_1)&=3D(\alpha a_1+ \alpha c_1 )x^2+(\alpha = a_1 -\alpha b_1 )x+(\alpha b_1 +\alpha c_1 )\\ % &=3D\alpha \Big(a_1+ c_1 )x^2+(a_1 -b_1 )x+b_1 +c_1\Big)\\ % &=3D\alpha T(\u_1). %\end{align*} %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %$T$=20 %=C3=A4=C3=B2=C3=BA=C3=B7=C3=A4 =C3=AC=C3=A9=C3=B0=C3=A0=C3=B8=C3=A9=C3=BA. \item=20 =C3=A4=C3=B8=C3=A0=C3=A5 =C3=AB=C3=A9=20 $x^2+1\in \ip(T)$ =C3=A5=C3=A2=C3=AD=20 $-x+1\in \ip(T)$. %\answer=20 % %=C3=B0=C3=B7=C3=A7=20 %$a=3Db=3D1, c=3D0$=20 %=C3=A5=C3=B0=C3=B7=C3=A1=C3=AC=20 %$T\bm 1\\1\\0 \em=3D(1+0)x^2+(1-1)x+1+0=3Dx^2+1$=20 %=C3=AC=C3=AB=C3=AF=20 %$x^2+1\in \ip(T)$. % %=C3=B0=C3=B7=C3=A7=20 %$a=3Dc=3D0, b=3D1$=20 %=C3=A5=C3=B0=C3=B7=C3=A1=C3=AC=20 %$T\bm 0\\0\\1 \em=3D(0+0)x^2+(0-1)x+1+0=3D-x+1$=20 %=C3=AC=C3=AB=C3=AF=20 %$-x+1\in \ip(T)$. \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=AB=C3=A9=20 $\dim (\ip(T))\geq 2$. %\answer=20 % %=C3=A1=C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=B7=C3=A5=C3=A3=C3=AD =C3=B8=C3=A0= =C3=A9=C3=B0=C3=A5 =C3=AB=C3=A9=20 %$x^2+1, -x+1\in \ip(T)$.=20 %=C3=A1=C3=B8=C3=A5=C3=B8 =C3=B9=C3=A4=C3=AD =C3=A1=C3=BA"=C3=AC =C3=AB=C3= =A9 =C3=A4=C3=AE=C3=B2=C3=AC=C3=A5=C3=BA =C3=B9=C3=A5=C3=B0=C3=A5=C3=BA. %=C3=AE=C3=B6=C3=A3 =C3=B9=C3=B0=C3=A9,=20 %$\sp\{x^2+1, -x+1\}\subseteq \ip(T)$. % =C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %\[\dim (\ip(T))\geq \dim (\sp\{ x^2+1, -x+1 \})=3D2.\] \item=20 =C3=A4=C3=B8=C3=A0=C3=A5 =C3=AB=C3=A9=20 $\bm 1\\1\\-1 \em\in \ker(T)$. %\answer % %=C3=B0=C3=A7=C3=B9=C3=A1=20 %\[T\bm 1\\1\\-1 \em=3D(1-1)x^2+(1-1)x+1-1=3D0\] %=C3=AC=C3=AB=C3=AF=20 %$\bm 1\\1\\-1\em\in \ker T$. \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=AB=C3=A9=20 $\dim (\ker(T))\geq 1$. %\answer % %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AE=C3=A4=C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=B7= =C3=A5=C3=A3=C3=AD =C3=AB=C3=A9=20 %$\sp\left\{ \bm 1\\1\\-1 \em \right\}\subseteq \ker T$ %=C3=AC=C3=AB=C3=AF=20 %\[\dim(\ker T)\geq \dim \left(\sp\left\{ \bm 1\\1\\-1 \em \right\}\right)= =3D1.\] \item=20 =C3=A7=C3=B9=C3=A1=C3=A5 =C3=A0=C3=BA=20 $\dim (\ker (T))$=20 =C3=A5=C3=A0=C3=BA=20 $\dim (\ip(T))$. %\answer % %=C3=AC=C3=B4=C3=A9 =C3=AE=C3=B9=C3=B4=C3=A8 =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9= =C3=AD=20 %$\dim (\ker (T))+\dim (\ip(T))=3D\dim(\setR^3)=3D3$.=20 %=C3=AC=C3=B4=C3=A9 =C3=A4=C3=B1=C3=B2=C3=A9=C3=B4=C3=A9=C3=AD =C3=B7=C3=A5= =C3=A3=C3=AE=C3=A9=C3=AD=20 %$\dim (\ker (T))\geq 1$=20 %=C3=A5=20 %$\dim (\ip(T))\geq 2$ %=C3=AC=C3=AB=C3=AF =C3=A1=C3=A4=C3=AB=C3=B8=C3=A7=20 %$\dim (\ker (T))=3D 1$=20 %=C3=A5=20 %$\dim (\ip(T))=3D 2$. %\item=20 %=C3=A9=C3=A4=C3=A9=20 %$B=3D\{x^2, x, 1\}$ %=C3=A4=C3=A1=C3=B1=C3=A9=C3=B1 =C3=A4=C3=B1=C3=A8=C3=B0=C3=A3=C3=B8=C3=A8= =C3=A9 =C3=B9=C3=AC=20 %$V$. %=C3=BA=C3=A4=C3=A9=20=20 %$A\in \setR^{3\times 3}$ %=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=AE=C3=B7=C3=A9=C3=A9=C3=AE= =C3=BA=20 %$[T(\u)]_B=3DA\u$=20 %=C3=AC=C3=AB=C3=AC=20 %$\u\in \setR^3$. %\begin{enumerate} % \item=20 % =C3=AE=C3=B6=C3=A0=C3=A5 =C3=A0=C3=BA=20 % $A$=20 % =C3=A1=C3=AE=C3=B4=C3=A5=C3=B8=C3=B9. %=20 %%\answer %% %%=C3=B0=C3=AB=C3=BA=C3=A5=C3=A1=20 %%$\u=3D\bm a\\b\\c \em$ %%=C3=A5=C3=B0=C3=A7=C3=B9=C3=A1=20 %%\[[T\u]_B=3D[(a+c)x^2+(a-b)x+(b+c)]_B=3D\bm a+c\\ a-b\\ b+c \em.\] %%=C3=B0=C3=A3=C3=B8=C3=B9 =C3=AC=C3=AE=C3=B6=C3=A5=C3=A0 =C3=AE=C3=A8=C3= =B8=C3=A9=C3=B6=C3=A4=20 %%$A$=20 %%=C3=AB=C3=AA =C3=B9=20 %%$A\u=3DA\bm a \\ b \\ c \em=3D\bm a+c\\ a-b\\ b+c \em$. %%=C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=B7=C3=A7=C3=BA=20 %%\[A=3D\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\] %%=C3=AB=C3=A9 =C3=A0=C3=A6=20 %%\[A\bm a \\ b \\ c \em=3D\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\bm a \\ b \\ c \e= m=3D\bm a+c\\ a-b\\ b+c \em.\] % %\item %=C3=A4=C3=B8=C3=A0=C3=A5 =C3=AB=C3=A9=20 %$\Nu(A)=3D\sp\left\{ \bm 1\\1\\-1 \em\right\}$. % %%\answer %% %% %%=C3=AB=C3=B2=C3=BA,=20 %%$\Nu(A)$=20 %%=C3=A4=C3=A5=C3=A0 =C3=A0=C3=A5=C3=B1=C3=B3 =C3=A4=C3=B4=C3=BA=C3=B8=C3= =A5=C3=B0=C3=A5=C3=BA =C3=AC=C3=AE=C3=AE"=C3=AC=20 %%$A\x=3D\0$.=20 %%=C3=B0=C3=B4=C3=BA=C3=A5=C3=B8:=20 %%\begin{align*} %%(A|\0)=3D\bse{ccc}=20 %%1&0&1&0\\ 1&-1&0&0\\ 0&1&1&0 \ese & %%\widesim{R_2\to R_2-R_1}=20 %%\bse{ccc}=20 %%1&0&1&0\\ 0&-1&-1&0\\ 0&1&1&0 \ese\\ %%&\widesim{R_3\to R_3+R_2}=20 %%\bse{ccc}=20 %%1&0&1&0\\ 0&-1&-1&0\\ 0&0&0&0 \ese %%\widesim{R_2\to -R_2}=20 %%\bse{ccc}=20 %%1&0&1&0\\ 0&1&1&0\\ 0&0&0&0 \ese. %%\end{align*} %% %%=C3=B0=C3=AB=C3=BA=C3=A5=C3=A1=20 %%$c=3Dt$=20 %%=C3=A5=C3=A0=C3=A6 =C3=B0=C3=B7=C3=A1=C3=AC=20 %%$a=3Db=3D-t$=20 %%=C3=AC=C3=AB=C3=AF =C3=A9=C3=B9 =C3=B4=C3=BA=C3=B8=C3=A5=C3=AF =C3=AB=C3= =AC=C3=AC=C3=A9=20 %%$\bm a\\ b\\ c\em=3D\left\{ t\bm -1\\-1\\1 \em\mid t\in \setR \right\}$ %%=C3=AC=C3=AB=C3=AF=20 %%\[\Nu(A)=3D\sp \left\{\bm -1\\-1\\1 \em\right\}=3D\sp \left\{\bm 1\\1\\-1= \em\right\}.\] % %\item=20 %=C3=AE=C3=B6=C3=A0=C3=A5 =C3=A1=C3=B1=C3=A9=C3=B1 =C3=AC=20 %$\C(A)$. % %%\answer %% %%=C3=AC=C3=B4=C3=A9 =C3=A4=C3=A4=C3=A2=C3=A3=C3=B8=C3=A4 =C3=B9=C3=AC =C3= =AE=C3=B8=C3=A7=C3=A1 =C3=B2=C3=AE=C3=A5=C3=A3=C3=A5=C3=BA %%$\C(A)=3D\sp\left\{ %%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em, \bm 1\\0\\1 \em %%\right\}.$ %%=C3=AB=C3=A1=C3=B8 =C3=B8=C3=A0=C3=A9=C3=B0=C3=A5 =C3=AB=C3=A9=20 %%$\bm 1\\ 1\\ -1\em\in \Nu(A)$=20 %%=C3=AC=C3=AB=C3=AF=20 %%\[\bm 0\\0\\0 \em=3D\bm 1&0&1\\ 1&-1&0\\ 0&1&1 \em\bm 1\\ 1\\ -1\em=3D1\b= m 1\\1\\0\em+1\bm 0\\ -1\\ 1 \em+(-1)\bm 1\\0\\1 \em.\] %%=C3=AC=C3=AB=C3=AF =C3=B0=C3=A5=C3=AB=C3=AC =C3=AC=C3=AB=C3=BA=C3=A5=C3= =A1 =C3=A0=C3=BA =C3=A4=C3=B2=C3=AE=C3=A5=C3=A3=C3=A4 =C3=A4=C3=A0=C3=A7=C3= =B8=C3=A5=C3=B0=C3=A4 =C3=AB=C3=B6=C3=A9=C3=B8=C3=A5=C3=B3 =C3=AC=C3=A9=C3= =B0=C3=A0=C3=B8=C3=A9 =C3=B9=C3=AC =C3=A4=C3=B2=C3=AE=C3=A5=C3=A3=C3=A5=C3= =BA =C3=A4=C3=A0=C3=A7=C3=B8=C3=A5=C3=BA:=20 %%\[\bm 1\\0\\1 \em=3D\bm 1\\1\\0\em+\bm 0\\ -1\\ 1 \em.\] %%=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %%$\C(A)=3D\sp\left\{ %%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em, \bm 1\\0\\1 \em %%\right\}=3D\sp\left\{ %%\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em %%\right\}.$ %%=C3=A4=C3=A5=C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=C3=A9=C3=AD=20 %%$\bm 1\\1\\0\em, \bm 0\\ -1\\ 1 \em$=20 %%=C3=A4=C3=AD =C3=A1=C3=BA"=C3=AC =C3=AB=C3=A9 =C3=A4=C3=AD =C3=A0=C3=A9= =C3=B0=C3=AD =C3=AE=C3=B7=C3=A1=C3=A9=C3=AC=C3=A9=C3=AD =C3=AC=C3=AB=C3=AF = =C3=A4=C3=AD =C3=AE=C3=A4=C3=A5=C3=A5=C3=A9=C3=AD =C3=A1=C3=B1=C3=A9=C3=B1 = =C3=AC=20 %%$\C(A)$.=20 %\end{enumerate} \end{enumerate} \item=20 =C3=A9=C3=A4=C3=A9=20 $\setF$=20 =C3=B9=C3=A3=C3=A4.=20 \begin{enumerate} \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=A0=C3=AD=20 $A, B\in \setF^{n\times n}$=20 =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A5=C3=BA =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8= =C3=A9=C3=A5=C3=BA =C3=A0=C3=A6=C3=A9 $AB$=20 =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA =C3=A0=C3=AD =C3=A5=C3=B8=C3=B7 = =C3=A0=C3=AD=20 $AB=3DBA$. =09=09=20 %\answer=20 % %=C3=B0=C3=BA=C3=A5=C3=AF =C3=B9=20 %$A, B$=20 %=C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=A5=C3=BA =C3=AC=C3=AB=C3=AF=20 %$A^T=3DA, B^T=3DB$.=20 %=C3=AB=C3=B2=C3=BA,=20 %$(AB)^T=3DB^TA^T=3DBA$.=20 %=C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=A4=C3=B1=C3=A9=C3=B7 =C3=AB=C3=A9=20 %$AB$ %=C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA =C3=A0=C3=AD =C3=A5=C3=B8=C3=B7= =C3=A0=C3=AD=20 %$(AB)^T=3DAB$=20 %=C3=A0=C3=AD =C3=A5=C3=B8=C3=B7 =C3=A0=C3=AD=20 %$BA=3DAB$.=09=09 =09=09 \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=A0=C3=AD=20 $A, B\in \setR^{3\times 3}$ =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A5=C3=BA =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8= =C3=A9=C3=A5=C3=BA=20 =C3=AB=C3=AA =C3=B9=20 $AB=3D\bm 2&0&0\\ 0&5&0\\ 0&0&-1 \em$ =C3=A0=C3=A6=C3=A9 =C3=A2=C3=AD=20 $BA=3D\bm 2&0&0\\ 0&5&0\\ 0&0&-1 \em$. %\answer % %=C3=A4=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 %$AB=3D\bm 2&0&0\\ 0&5&0\\ 0&0&-1 \em$ %=C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA =C3=AB=C3=A9 =C3=A4=C3=A9=C3=A0= =C3=A0=C3=AC=C3=AB=C3=B1=C3=A5=C3=B0=C3=A9=C3=BA. =C3=AC=C3=B4=C3=A9 =C3= =A4=C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=B7=C3=A5=C3=A3=C3=AD =C3=B0=C3=A5=C3= =A1=C3=B2 =C3=AB=C3=A9=20 %$BA=3DAB=3D\bm 2&0&0\\ 0&5&0\\ 0&0&-1 \em$. \item=20 =C3=BA=C3=B0=C3=A5 =C3=A3=C3=A5=C3=A2=C3=AE=C3=A4 =C3=AC=C3=AE=C3=A8=C3=B8= =C3=A9=C3=B6=C3=A5=C3=BA=20 $A, B\in \setR^{2\times 2}$ )=C3=AC=C3=A0 =C3=A1=C3=A4=C3=AB=C3=B8=C3=A7 =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8= =C3=A9=C3=A5=C3=BA( =C3=AB=C3=AA =C3=B9=20 $AB$ =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA =C3=A5 ${AB\neq BA}$.=09=09 %\answer=20 % %=C3=B0=C3=B7=C3=A7=20 %$A=3D\bm 1&1\\0&1 \em, B=3D\bm 1&0\\ 1&1 \em$. %=C3=A0=C3=A6 %$AB=3D\bm 1&1\\0&1 \em \bm 1&0\\ 1&1 \em=3D\bm 2&1\\1&1 \em$ %=C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA =C3=A5=20 %$BA=3D\bm 1&0\\ 1&1 \em \bm 1&1\\0&1 \em =3D \bm 1&1\\1&2 \em \neq AB$. \end{enumerate} \item \label{inverse-of-products} =C3=A9=C3=A4=C3=A9=20 $\setF$ =C3=B9=C3=A3=C3=A4 =C3=A5=C3=BA=C3=A4=C3=A9=C3=A9=C3=B0=C3=A4=20 $A, B\in \setF^{n\times n}$ =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A5=C3=BA =C3=A4=C3=B4=C3=A9=C3=AB=C3=A5= =C3=BA.=20 \begin{enumerate} \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=20 $AB$=20 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. %\answer % %=C3=AC=C3=B4=C3=A9 =C3=AE=C3=B9=C3=B4=C3=A8=20 %$\det (AB)=3D\det A\det B$.=20 %=C3=AE=C3=AB=C3=A9=C3=A5=C3=A5=C3=AF =C3=B9=20 %$A, B$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A5=C3=BA =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD= =20 %$\det A, \det B\neq 0$.=20 %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %$\det (AB)\neq 0$=20 %=C3=AC=C3=AB=C3=AF=20 %$AB$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=AB=C3=A9=20 $(AB)^{-1}=3DB^{-1}A^{-1}$. =09=09 %\answer=09=09 %=09=09 %=C3=B2=C3=AC =C3=AE=C3=B0=C3=BA =C3=AC=C3=A4=C3=B8=C3=A0=C3=A5=C3=BA =C3= =AB=C3=A9=20 %$B^{-1}A^{-1}$=20 %=C3=A4=C3=A4=C3=A5=C3=B4=C3=AB=C3=A9=C3=BA =C3=B9=C3=AC $AB$ =C3=A0=C3=B0= =C3=A5 =C3=B0=C3=A7=C3=B9=C3=A1:=20 %\[ (AB)(B^{-1}A^{-1})=3D A(BB^{-1})A^{-1}=3DAIA^{-1}=3DAA^{-1}=3DI. \]=09= =09 %=C3=AC=C3=B4=C3=A9 =C3=AE=C3=B9=C3=B4=C3=A8, =C3=AE=C3=BA=C3=B7=C3=A9=C3= =A9=C3=AD =C3=A2=C3=AD=20 %$(B^{-1}A^{-1})(AB)=3DI$ %=C3=AC=C3=AB=C3=AF=20 %$B^{-1}A^{-1}$ %=C3=A0=C3=AB=C3=AF =C3=A4=C3=A4=C3=A5=C3=B4=C3=AB=C3=A9=C3=BA =C3=B9=C3=AC= =20 %$AB$. \item=20 =C3=A0=C3=AD=20 $C\in \setF^{n\times n}$ =C3=A2=C3=AD =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4, =C3=AE=C3=B6=C3=A0=C3=A5 =C3= =B0=C3=A5=C3=B1=C3=A7=C3=A4 =C3=B2=C3=A1=C3=A5=C3=B8=20 $(ABC)^{-1}$ =C3=B9=C3=A1=C3=A4 =C3=AE=C3=A5=C3=B4=C3=A9=C3=B2=C3=A9=C3=AD =C3=A4=C3=AE= =C3=A8=C3=B8=C3=A9=C3=B6=C3=A5=C3=BA=20 $A^{-1}, B^{-1}, C^{-1}$. %\answer=20 % %=C3=AC=C3=B4=C3=A9 =C3=A4=C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=B7=C3=A5=C3=A3= =C3=AD =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD=20 %$(ABC)^{-1}=3D((AB)C)^{-1}=3DC^{-1}(AB)^{-1}=3DC^{-1}B^{-1}A^{-1}$. %=09=09 \item=20=09=09 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=20 $A^2$=20 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. =09=09 %\answer=20 % %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=A9=C3=B9=C3=A9=C3=B8=C3=A5=C3=BA =C3=AE=C3=B1= =C3=B2=C3=A9=C3=B3 =C3=A0' =C3=AB=C3=A0=C3=B9=C3=B8=20 %$B=3DA$ %=C3=AB=C3=A9 =C3=A0=C3=A6 %$AB=3DA^2$.=09=09 =09=09 \item =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9 $(A^2)^{-1}=3D(A^{-1})^2$. =09=09 %\answer=09 %=09=09 %=09=09 % =C3=B0=C3=B7=C3=A7=20 % $B=3DA$=20 % =C3=A5=C3=A0=C3=A6 =C3=AC=C3=B4=C3=A9 =C3=B1=C3=B2=C3=A9=C3=B3 =C3=B9=C3= =AC =C3=A4=C3=B9=C3=A0=C3=AC=C3=A4 =C3=A4=C3=B0=C3=A5=C3=AB=C3=A7=C3=A9=C3= =BA =C3=A1' =C3=B0=C3=B7=C3=A1=C3=AC % \[(A^2)^{-1}=3D(AA)^{-1}=3D(A^{-1})(A^{-1})=3D(A^{-1})^2.\] \item=20 =C3=A0=C3=AD=20 \[A=3D\bm -11&3&-4\\ -18&5&-7\\ 8&-2&3 \em, B=3D\bm 1&-1&1\\1&2&5\\ -2&4&= 1 \em\] =C3=A7=C3=B9=C3=A1=C3=A5 =C3=A0=C3=BA=20 $AB$=20 =C3=A5=C3=A0=C3=BA=20 $B^{-1}A^{-1}$. =C3=AE=C3=A5=C3=AE=C3=AC=C3=B5 =C3=AC=C3=A4=C3=A9=C3=B2=C3=A6=C3=B8 =C3=A1= =C3=B1=C3=B2=C3=A9=C3=B3 =C3=A1' =C3=A1=C3=AE=C3=B7=C3=A5=C3=AD =C3=AC=C3= =A7=C3=B9=C3=A1 =C3=A0=C3=BA=20 $A^{-1}$=20 =C3=A5=C3=A0=C3=BA=20 $B^{-1}$=20 =C3=A1=C3=B0=C3=B4=C3=B8=C3=A3. =09=09 %\answer=09=09 %=09=09 %\[AB=3D\bm 0&1&0\\1&0&0\\0&0&1 \em.\]=09=09 %=C3=AC=C3=B4=C3=A9 =C3=B1=C3=B2=C3=A9=C3=B3 =C3=A1' =C3=AE=C3=BA=C3=B7=C3= =A9=C3=A9=C3=AD=20 %$B^{-1}A^{-1}=3D(AB)^{-1}=3D\bm 0&1&0\\1&0&0\\0&0&1 \em^{-1}$.=09 %=C3=B0=C3=A7=C3=B9=C3=A1 =C3=A0=C3=BA =C3=A4=C3=A4=C3=A5=C3=B4=C3=AB=C3=A9= =C3=BA: %\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}=20=09 % \right)=20=09 % &{\widesim{R_1\leftrightarrow R_2}}=09 % \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}=20=09 % \right). %\end{align*} % =C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 % $B^{-1}A^{-1}=3D\bm 0&1&0\\1&0&0\\0&0&1 \em^{-1}=3D\bm 0&1&0\\1&0&0\\0&0&= 1 \em$. =09 \end{enumerate}=09 =09 =09 \item \label{eigen-of-inverse} =C3=A9=C3=A4=C3=A9=20 $\setF$ =C3=B9=C3=A3=C3=A4 =C3=A5=C3=BA=C3=A4=C3=A9=20 $A\in \setF^{n\times n}$ =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4.=20 \begin{enumerate} \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=AB=C3=AC =C3=B2=C3=B8=C3=AA = =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 $A$=20 =C3=B9=C3=A5=C3=B0=C3=A4 =C3=AE=C3=A0=C3=B4=C3=B1. %\answer % %=C3=B0=C3=BA=C3=A5=C3=AF =C3=B9=20 %$A$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. =C3=AC=C3=AB=C3=AF, =C3=AC=C3=B4=C3=A9 =C3= =A8=C3=B2=C3=B0=C3=A4,=20 %$\Nu(A)=3D\{\0\}$.=20 %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=B9=20 %$A\x\neq \0$=20 %=C3=AC=C3=AB=C3=AC=20 %$\x\neq \0$. %=C3=AC=C3=AB=C3=AF =C3=AC=C3=A0 =C3=A9=C3=BA=C3=AB=C3=AF =C3=B9=20 %$A\x=3D0\x$=20 %=C3=B2=C3=A1=C3=A5=C3=B8 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=20 %$\x\neq \0$. %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=B9=20 %$0$=20 %=C3=A0=C3=A9=C3=B0=C3=A5 =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9 =C3= =B9=C3=AC=20 %$A$. \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=A0=C3=AD=20 $\x$=20 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 $A$=20 =C3=B2=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 $\lambda$=20 =C3=A0=C3=A6=C3=A9=20 $\x$=20 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 $A^{-1}$ =C3=B2=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 $\frac{1}{\lambda}$. =C3=AB=C3=A0=C3=AF=20 $A^{-1}$=20 =C3=AE=C3=B1=C3=AE=C3=AF =C3=A0=C3=BA =C3=A4=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6= =C3=A4 =C3=A4=C3=A4=C3=A5=C3=B4=C3=AB=C3=A9=C3=BA =C3=B9=C3=AC $A$. %\answer % %=C3=AC=C3=B4=C3=A9 =C3=A4=C3=B0=C3=BA=C3=A5=C3=AF =C3=AE=C3=BA=C3=B7=C3=A9= =C3=A9=C3=AD=20 %$A\x=3D\lambda \x$.=20 %=C3=B0=C3=AB=C3=B4=C3=A9=C3=AC =C3=A0=C3=BA =C3=B9=C3=B0=C3=A9 =C3=A4=C3= =A0=C3=A2=C3=B4=C3=A9=C3=AD =C3=A1=20 %$A^{-1}$=20 %=C3=A5=C3=B0=C3=B7=C3=A1=C3=AC=20 %\[A^{-1}A\x=3DA^{-1}(\lambda \x)=3D\lambda A^{-1}\x.\] %=C3=AB=C3=B2=C3=BA, =C3=AC=C3=B4=C3=A9 =C3=A4=C3=A4=C3=A2=C3=A3=C3=B8=C3= =A4 =C3=B9=C3=AC =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=A5=C3=B4=C3= =AB=C3=A9=C3=BA =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD=20 %$A^{-1}A=3DI$=20 %=C3=AB=C3=A0=C3=B9=C3=B8=20 %$I$=20 %=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=BA =C3=A4=C3=A9=C3=A7=C3=A9=C3=A3=C3=A4 = =C3=AE=C3=B1=C3=A3=C3=B8=20 %$n\times n$. %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=B9=20 %$A^{-1}A\x=3DI\x=3D\x$=20 %=C3=AC=C3=AB=C3=AF=20 %$\x=3D\lambda A^{-1}\x$. %=C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=A4=C3=B1=C3=A9=C3=B7 =C3=AB=C3=A9=20 %$A^{-1}\x=3D\frac{1}{\lambda}\x$ %=C3=AC=C3=AB=C3=AF=20 %$\x$=20 %=C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 %$A^{-1}$=20 %=C3=B2=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 %$\frac{1}{\lambda}$. \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=A0=C3=AD=20 $\x$=20 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 $A$=20 =C3=B2=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 $\lambda$=20 =C3=A0=C3=A6=C3=A9=20 $\x$=20 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 $A^2$ =C3=B2=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 $\lambda^2$. %\answer % %=C3=AC=C3=B4=C3=A9 =C3=A4=C3=B0=C3=BA=C3=A5=C3=AF=20 %$A\x=3D\lambda\x$. %=C3=AB=C3=B2=C3=BA, %$A^2\x=3DAA\x=3DA\lambda \x=3D\lambda A\x=3D\lambda (\lambda \x)=3D\lambda= ^2\x$ %=C3=AC=C3=AB=C3=AF=20 %$\x$=20 %=C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 %$A^2$=20 %=C3=B2=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 %$\lambda^2$. \end{enumerate} \item=20 =C3=A9=C3=A4=C3=A9=20 $\setF$ =C3=B9=C3=A3=C3=A4 =C3=A5=C3=BA=C3=A4=C3=A9=20 $A\in \setF^{n\times n}$ =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4.=20 \begin{enumerate} \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=AB=C3=A9=20 $(A^T)^{-1}=3D(A^{-1})^T$.=09 %\answer % %=C3=AB=C3=A3=C3=A9 =C3=AC=C3=A1=C3=A3=C3=A5=C3=B7 =C3=A4=C3=A0=C3=AD=20 %$(A^{-1})^T$ %=C3=A4=C3=A4=C3=A5=C3=B4=C3=AB=C3=A9=C3=BA =C3=B9=C3=AC=20 %$A^T$, %=C3=B0=C3=A7=C3=B9=C3=A1 =C3=A0=C3=BA=20 %$A^T(A^{-1})^T$.=20 %=C3=AB=C3=B2=C3=BA,=20 %\begin{align*} %A^T(A^{-1})^T&=3D(A^{-1}A)^T\\ %&=3DI^T=3DI. %\end{align*} %=C3=AC=C3=B4=C3=A9 =C3=AE=C3=B9=C3=B4=C3=A8 =C3=B0=C3=A5=C3=A1=C3=B2 =C3= =B9=C3=A2=C3=AD=20 %$(A^{-1})^TA^T=3DI$. %=C3=AC=C3=AB=C3=AF =C3=A0=C3=AB=C3=AF=20 %$(A^{-1})^T$ %=C3=A4=C3=A4=C3=A5=C3=B4=C3=AB=C3=A9=C3=BA =C3=B9=C3=AC=20 %$A^T$. \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=A0=C3=AD=20 $A$=20 =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA =C3=A0=C3=A6=C3=A9 =C3=A2=C3=AD= =20 $A^{-1}$ =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA. %\answer % %=C3=A0=C3=AD $A$ =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA =C3=A0=C3=A6=20 %$A^T=3DA$.=20 %=C3=AC=C3=B4=C3=A9 =C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=B7=C3=A5=C3=A3=C3=AD= =C3=B0=C3=B7=C3=A1=C3=AC=20 %\[(A^{-1})^T=3D(A^T)^{-1}=3DA^{-1}\] %=C3=AC=C3=AB=C3=AF=20 %$A^{-1}$ %=C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9=C3=BA. \item=20 =C3=A0=C3=AD=20 \[A=3D\bm 3&0&5\\0&2&1\\5&1&9 \em\] \begin{enumerate} \item =C3=A7=C3=B9=C3=A1=C3=A5 =C3=A0=C3=BA=20 $A^{-1}$. %\answer % %=C3=B2=C3=AC =C3=AE=C3=B0=C3=BA =C3=AC=C3=A7=C3=B9=C3=A1 =C3=A0=C3=BA =C3= =A4=C3=A4=C3=A5=C3=AB=C3=B4=C3=A9=C3=BA =C3=B9=C3=AC $A$ =C3=B0=C3=A3=C3=B8= =C3=A2: % %\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}=20=09 % \right)=20=09 % &{\widesim{R_3\to 3R_3-5R_1}}=09 % \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}=20=09 % \right)\\[0.5em] % &{\widesim{R_3\to 2R_3-3R_2}}=09 % \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}=20=09 % \right)\\[0.5em] % &{\widesim{R_1\to R_1-5R_2}}=09 % \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}=20=09 % \right)\\[0.5em] % &{\widesim{R_2\to R_2-R_3}}=09 % \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}=20=09 % \right)\\[0.5em] % &{\widesim{R_1\to \frac{1}{3}R_1}}=09 % \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}=20=09 % \right)\\[0.5em]=09=09=09=09 % &{\widesim{R_2\to \frac{1}{2}R_2}}=09 % \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}=20=09 % \right)\\=09=09=09=09=09=09 %\end{align*} %=C3=AC=C3=AB=C3=AF=20 %$A^{-1}=3D\bm %17&5&-10\\ %5&2&-3\\ %-10&-3&6 %\em$. \item=20 =C3=A7=C3=B9=C3=A1=C3=A5 =C3=A0=C3=BA=20 $(A^2)^{-1}$. %\answer % %=C3=AC=C3=B4=C3=A9 =C3=B9=C3=A0=C3=AC=C3=A4=20 %\ref{inverse-of-products} % =C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4' =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD=20 %$(A^2)^{-1}=3D(A^{-1})^2$. %=C3=AC=C3=B4=C3=A9 =C3=A4=C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=B7=C3=A5=C3=A3= =C3=AD =C3=B0=C3=B7=C3=A1=C3=AC=20 %\begin{align*} %(A^2)^{-1}&=3D(A^{-1})^2\\ %&=3DA^{-1}A^{-1}\\ %&=3D\bm %17&5&-10\\ %5&2&-3\\ %-10&-3&6 %\em\bm %17&5&-10\\ %5&2&-3\\ %-10&-3&6 %\em\\ %&=3D\bm %414&125&-245\\ %125&38&-74\\ %-245&-74&145 %\em %\end{align*} \end{enumerate} \end{enumerate} \textbf{=C3=BA=C3=A6=C3=AB=C3=A5=C3=B8=C3=BA:} =C3=B2=C3=A1=C3=A5=C3=B8 =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=B4= =C3=A9=C3=AB=C3=A4 $\bm p&q\\ r&s \em$ =C3=AE=C3=B1=C3=A3=C3=B8=20 $2\times 2$ =C3=A4=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=A4=C3=A5=C3=B4=C3=AB= =C3=A9=C3=BA =C3=B0=C3=BA=C3=A5=C3=B0=C3=A4 =C3=B2=C3=AC =C3=A9=C3=A3=C3=A9= =20 \[\bm p&q\\ r&s \em^{-1}=3D\frac{1}{ps-qr}\bm s&-q\\ -r&p \em.\] \item=20 =C3=A9=C3=A4=C3=A9=C3=A5 $a, b\in \setR$=20 =C3=AB=C3=AA =C3=B9=C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD=20 $a^2-b^2=3D1$. =C3=B0=C3=A2=C3=A3=C3=A9=C3=B8 =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A5=C3=BA= =20 \[C=3D\bm a&b\\ b&a \em, D=3D\bm a&b\\ a&b \em \in \setR^{2\times 2}.\] \begin{enumerate} \item=20 =C3=A4=C3=B8=C3=A0=C3=A5 =C3=B9 $D$ =C3=A0=C3=A9=C3=B0=C3=A4 =C3=A4=C3=B4= =C3=A9=C3=AB=C3=A4. %\answer=20 % %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 1:}=09 %$\det D=3Dab-ba=3D0$=20 %=C3=AC=C3=AB=C3=AF $D$ =C3=A0=C3=A9=C3=B0=C3=A4 =C3=A4=C3=B4=C3=A9=C3=AB= =C3=A4. % %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 2:} %=C3=A4=C3=B2=C3=AE=C3=A5=C3=A3=C3=A5=C3=BA =C3=B9=C3=AC $D$ =C3=AE=C3=B7= =C3=A1=C3=A9=C3=AC=C3=A5=C3=BA =C3=AC=C3=AB=C3=AF =C3=A4=C3=AF =C3=BA=C3=AC= =C3=A5=C3=A9=C3=A5=C3=BA =C3=AC=C3=A9=C3=B0=C3=A0=C3=B8=C3=A9=C3=BA. =C3=B0= =C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %$D$=20 %=C3=A0=C3=A9=C3=B0=C3=A4 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. % % %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 3:} %=C3=A4=C3=B9=C3=A5=C3=B8=C3=A5=C3=BA =C3=B9=C3=AC $D$ =C3=A6=C3=A4=C3=A5= =C3=BA =C3=AC=C3=AB=C3=AF=20 %$\r (D)<2$=20 %)=C3=A1=C3=B2=C3=B6=C3=AD=20 %$\r (D)=3D1$=20 %=C3=AB=C3=A9=20 %$a^2-b^2=3D1$=20 %=C3=AC=C3=AB=C3=AF=20 %$(a, b)\neq (0,0)$.( %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %$D$=20 %=C3=A0=C3=A9=C3=B0=C3=A4 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. % %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 4:} %$D\bm b\\-a \em=3D\bm 0\\0 \em$ %=C3=AC=C3=AB=C3=AF=20 %$\bm b\\ -a\em\in \Nu(D)$. %=C3=A0=C3=A1=C3=AC=20 %$\bm b\\-a \em\neq \bm 0\\0 \em$ %=C3=AB=C3=A9=20 %$a^2-b^2=3D1$. %=C3=B0=C3=A5=C3=A1=C3=B2 =C3=AB=C3=A9=20 %$\Nu(D)\neq \{\0\}$=20 %=C3=AC=C3=AB=C3=AF=20 %$D$=20 %=C3=A0=C3=A9=C3=B0=C3=A4 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. =09 \item=20 =C3=A4=C3=B8=C3=A0=C3=A5 =C3=AB=C3=A9=20 $C$=20 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4 =C3=A5=C3=AE=C3=B6=C3=A0=C3=A5 =C3=A0=C3=BA= =C3=A4=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=A4=C3=A4=C3=A5=C3=B4=C3=AB= =C3=A9=C3=BA=20 $C^{-1}$. =09 %\answer=09 %=09 %=C3=B0=C3=A7=C3=B9=C3=A1:=20 %$\det C=3Da^2-b^2=3D1\neq 0$=20 %=C3=AC=C3=AB=C3=AF=20 %$C$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. =C3=AC=C3=B4=C3=A9 =C3=A8=C3=B2=C3=B0=C3= =A4 =C3=A4=C3=A4=C3=A5=C3=B4=C3=AB=C3=A9=C3=BA =C3=B0=C3=BA=C3=A5=C3=B0=C3= =A4 =C3=B2=C3=AC =C3=A9=C3=A3=C3=A9=20 %$C^{-1}=3D\frac{1}{\det C}\bm a&-b\\ -b&a \em=3D\bm a&-b\\ -b&a \em$. =09 \item=20 =C3=AE=C3=B6=C3=A0=C3=A5 =C3=A0=C3=BA =C3=AB=C3=AC =C3=A4=C3=B2=C3=B8=C3= =AB=C3=A9=C3=AD =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD =C3=B9=C3=AC=20 $C$ )=C3=A9=C3=B9 =C3=AC=C3=AB=C3=BA=C3=A5=C3=A1 =C3=A0=C3=BA =C3=A4=C3=B2"=C3= =B2 =C3=A1=C3=A0=C3=AE=C3=B6=C3=B2=C3=A5=C3=BA =C3=A4=C3=B4=C3=B8=C3=AE=C3= =A8=C3=B8=C3=A9=C3=AD $a, b$.(=09 =C3=B2=C3=A1=C3=A5=C3=B8 =C3=AB=C3=AC =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3= =AE=C3=A9 =C3=AE=C3=B6=C3=A0=C3=A5 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3= =B6=C3=AE=C3=A9 =C3=AE=C3=BA=C3=A0=C3=A9=C3=AD. =09 %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 1:} %=C3=AC=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4=20 %$C$=20 %=C3=A9=C3=B9 =C3=A4=C3=B8=C3=A1=C3=A4 =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9= =C3=A4. =C3=A1=C3=B4=C3=B8=C3=A8, =C3=B1=C3=AB=C3=A5=C3=AD =C3=A4=C3=AE=C3= =B7=C3=A3=C3=AE=C3=A9=C3=AD =C3=A1=C3=AB=C3=AC =C3=B9=C3=A5=C3=B8=C3=A4 =C3= =B9=C3=A5=C3=A5=C3=A4. =C3=B0=C3=A5=C3=A1=C3=B2 =C3=B9=20 %$\bm 1\\ 1\em$=20 %=C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 %$C$=20 %=C3=A0=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 %$a+b$: % %\[C\bm 1\\ 1\em=3D\bm a&b\\ b&a \em\bm 1\\1 \em=3D\bm a+b\\ a+b \em=3D(a+b= )\bm 1\\1 \em.\]=09 %=C3=A1=C3=A0=C3=A5=C3=B4=C3=AF =C3=A3=C3=A5=C3=AE=C3=A4, =C3=A4=C3=B4=C3= =B8=C3=B9 =C3=A4=C3=AE=C3=B7=C3=A3=C3=AE=C3=A9=C3=AD =C3=A1=C3=AB=C3=AC =C3= =B9=C3=A5=C3=B8=C3=A4 =C3=A3=C3=A5=C3=AE=C3=A4 =C3=AC=C3=AB=C3=AF =C3=B0=C3= =A1=C3=A3=C3=A5=C3=B7 =C3=A0=C3=BA =C3=A4=C3=A5=C3=A5=C3=B7=C3=A8=C3=A5=C3= =B8=20 %$\bm 1\\ -1\em$: % %\[C\bm 1\\ -1\em=3D\bm a&b\\ b&a \em\bm 1\\-1 \em=3D\bm a-b\\ b-a \em=3D(a= -b)\bm 1\\-1 \em.\] %=C3=AC=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=AE=C3=B1=C3=A3=C3=B8=20 %$2\times 2$=20 %=C3=A9=C3=B9 =C3=AC=C3=AB=C3=AC =C3=A4=C3=A9=C3=A5=C3=BA=C3=B8 =C3=B9=C3= =B0=C3=A9 =C3=B9=C3=A5=C3=B8=C3=B9=C3=A9=C3=AD =C3=AC=C3=B4=C3=A5=C3=AC=C3= =A9=C3=B0=C3=A5=C3=AD =C3=A4=C3=A0=C3=A5=C3=B4=C3=A9=C3=A9=C3=B0=C3=A9 =C3= =AC=C3=AB=C3=AF =C3=A9=C3=B9 =C3=AC=C3=AB=C3=AC =C3=A4=C3=A9=C3=A5=C3=BA=C3= =B8 =C3=B9=C3=B0=C3=A9 =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3=B6=C3=AE=C3= =A9=C3=A9=C3=AD. =C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=A4=C3=B1=C3=A9=C3=B7 = =C3=B9=C3=AE=C3=B6=C3=A0=C3=B0=C3=A5 =C3=A0=C3=BA =C3=AB=C3=AC =C3=A4=C3=B2= =C3=B8=C3=AB=C3=A9=C3=AD =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD =C3=B9= =C3=AC $C$. %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 2:} %=C3=B0=C3=A7=C3=B9=C3=A1 =C3=A0=C3=BA =C3=A4=C3=B4=C3=A5=C3=AC=C3=A9=C3=B0= =C3=A5=C3=AD =C3=A4=C3=A0=C3=A5=C3=B4=C3=A9=C3=A9=C3=B0=C3=A9 =C3=B9=C3=AC = $C$: %\begin{align*} %p_C(\lambda)&=3D\bd a-\lambda&b\\ b&a-\lambda \ed\\ %&=3D(a-\lambda)^2-b^2\\ %&=3D\lambda^2-2a\lambda+a^2-b^2\\ %&=3D(\lambda-(a+b))(\lambda-(a-b))=09 %\end{align*}=09 %=C3=AC=C3=AB=C3=AF =C3=A9=C3=B9 =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3= =B6=C3=AE=C3=A9=C3=A9=C3=AD=20 %$a+b,\ \ a-b$. %=C3=B2=C3=AC =C3=AE=C3=B0=C3=BA =C3=AC=C3=AE=C3=B6=C3=A5=C3=A0 =C3=A5=C3= =B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B2=C3=A1=C3=A5=C3=B8 =C3= =B2=C3=B8=C3=AA =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=20 %$a+b$=20 %=C3=B0=C3=AE=C3=B6=C3=A0 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=AC=C3=A0 =C3= =A0=C3=B4=C3=B1 =C3=A1=20 %$\Nu(A-(a+b)I)$: %\begin{align*} %(A-(a+b)I|\0)&=3D\bse{cc} -b&b&0\\ b&-b&0 \ese{\widesim{R_2\to R_2+R_1}}\b= se{cc} -b&b&0\\ 0&0&0 \ese %\end{align*}=09 %=C3=A0=C3=A6 =C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=B7=C3=A7=C3=BA =C3=A5"=C3= =B2 %$\v_1=3D\bm1\\1 \em$.\\ % %=C3=B2=C3=AC =C3=AE=C3=B0=C3=BA =C3=AC=C3=AE=C3=B6=C3=A5=C3=A0 =C3=A5=C3= =B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B2=C3=A1=C3=A5=C3=B8 =C3= =B2=C3=B8=C3=AA =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=20 %$a-b$=20 %=C3=B0=C3=AE=C3=B6=C3=A0 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=AC=C3=A0 =C3= =A0=C3=B4=C3=B1 =C3=A1=20 %$\Nu(A-(a-b)I)$: %\begin{align*} % (A-(a-b)I|\0)&=3D\bse{cc} b&b&0\\ b&b&0 \ese{\widesim{R_2\to R_2-R_1}}\bs= e{cc} b&b&0\\ 0&0&0 \ese %\end{align*}=09 %=C3=A0=C3=A6 =C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=B7=C3=A7=C3=BA =C3=A5"=C3= =B2 %$\v_2=3D\bm1\\-1 \em$. \item=20 =C3=AE=C3=B6=C3=A0=C3=A5 =C3=A0=C3=BA =C3=AB=C3=AC =C3=A4=C3=B2=C3=B8=C3= =AB=C3=A9=C3=AD =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD =C3=B9=C3=AC=20 $D$ )=C3=A9=C3=B9 =C3=AC=C3=AB=C3=BA=C3=A5=C3=A1 =C3=A0=C3=BA =C3=A4=C3=B2"= =C3=B2 =C3=A1=C3=A0=C3=AE=C3=B6=C3=B2=C3=A5=C3=BA =C3=A4=C3=B4=C3=B8=C3=AE= =C3=A8=C3=B8=C3=A9=C3=AD $a, b$.( =C3=B2=C3=A1=C3=A5=C3=B8 =C3=AB=C3=AC =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3= =AE=C3=A9 =C3=AE=C3=B6=C3=A0=C3=A5 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3= =B6=C3=AE=C3=A9 =C3=AE=C3=BA=C3=A0=C3=A9=C3=AD. =09=09 %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 1:} %=C3=AB=C3=AE=C3=A5 =C3=A1=C3=B1=C3=B2=C3=A9=C3=B3 =C3=A4=C3=B7=C3=A5=C3=A3= =C3=AD, =C3=AC=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4=20 %$D$=20 %=C3=A9=C3=B9 =C3=A4=C3=B8=C3=A1=C3=A4 =C3=B1=C3=A9=C3=AE=C3=A8=C3=B8=C3=A9= =C3=A4. =C3=A1=C3=B4=C3=B8=C3=A8, =C3=B1=C3=AB=C3=A5=C3=AD =C3=A4=C3=AE=C3= =B7=C3=A3=C3=AE=C3=A9=C3=AD =C3=A1=C3=AB=C3=AC =C3=B9=C3=A5=C3=B8=C3=A4 =C3= =B9=C3=A5=C3=A5=C3=A4. =C3=B0=C3=A5=C3=A1=C3=B2 =C3=B9=20 %$\bm 1\\ 1\em$=20 %=C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B9=C3=AC=20 %$D$=20 %=C3=A0=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3=A9=20 %$a+b$: % %\[D\bm 1\\ 1\em=3D\bm a&b\\ a&b \em\bm 1\\1 \em=3D\bm a+b\\ a+b \em=3D(a+b= )\bm 1\\1 \em.\]=09 %=C3=AB=C3=B2=C3=BA, =C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=A0=C3=B4=C3=B1 =C3= =A0=C3=BA =C3=B9=C3=BA=C3=A9 =C3=A4=C3=B9=C3=A5=C3=B8=C3=A5=C3=BA =C3=A0=C3= =AD =C3=B0=C3=AB=C3=B4=C3=A9=C3=AC =C3=A1=C3=A5=C3=A5=C3=B7=C3=A8=C3=A5=C3= =B8 %$\bm b\\ -a\em$ %=C3=AC=C3=AB=C3=AF =C3=A4=C3=A5=C3=A0 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3= =B2=C3=B6=C3=AE=C3=A9 =C3=B2=C3=AD =C3=B2=C3=B8=C3=AA =C3=B2=C3=B6=C3=AE=C3= =A9=20 %$0$: % %\[D\bm b\\ -a\em=3D\bm a&b\\ a&b \em\bm b\\-a \em=3D\bm ab-ba\\ ab-ba \em= =3D\bm 0\\0 \em=3D0\bm b\\-a \em.\] %=C3=AC=C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4 =C3=AE=C3=B1=C3=A3=C3=B8=20 %$2\times 2$=20 %=C3=A9=C3=B9 =C3=AC=C3=AB=C3=AC =C3=A4=C3=A9=C3=A5=C3=BA=C3=B8 =C3=B9=C3= =B0=C3=A9 =C3=B9=C3=A5=C3=B8=C3=B9=C3=A9=C3=AD =C3=AC=C3=B4=C3=A5=C3=AC=C3= =A9=C3=B0=C3=A5=C3=AD =C3=A4=C3=A0=C3=A5=C3=B4=C3=A9=C3=A9=C3=B0=C3=A9 =C3= =AC=C3=AB=C3=AF =C3=A9=C3=B9 =C3=AC=C3=AB=C3=AC =C3=A4=C3=A9=C3=A5=C3=BA=C3= =B8 =C3=B9=C3=B0=C3=A9 =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3=B6=C3=AE=C3= =A9=C3=A9=C3=AD. =C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=A4=C3=B1=C3=A9=C3=B7 = =C3=B9=C3=AE=C3=B6=C3=A0=C3=B0=C3=A5 =C3=A0=C3=BA =C3=AB=C3=AC =C3=A4=C3=B2= =C3=B8=C3=AB=C3=A9=C3=AD =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD =C3=B9= =C3=AC $D$. %\textbf{=C3=B9=C3=A9=C3=A8=C3=A4 2:} %=C3=B0=C3=A7=C3=B9=C3=A1 =C3=A0=C3=BA =C3=A4=C3=B4=C3=A5=C3=AC=C3=A9=C3=B0= =C3=A5=C3=AD =C3=A4=C3=A0=C3=A5=C3=B4=C3=A9=C3=A9=C3=B0=C3=A9 =C3=B9=C3=AC = $D$: %\begin{align*} % p_D(\lambda)&=3D\bd a-\lambda&b\\ a&b-\lambda \ed\\ % &=3D(a-\lambda)(b-\lambda)-ab\\ % &=3D\lambda^2-(a+b)\lambda\\ % &=3D\lambda(\lambda-(a+b)) %\end{align*}=09 %=C3=AC=C3=AB=C3=AF =C3=A9=C3=B9 =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3= =B6=C3=AE=C3=A9=C3=A9=C3=AD=20 %$a+b,\ \ 0$. %=C3=B2=C3=AC =C3=AE=C3=B0=C3=BA =C3=AC=C3=AE=C3=B6=C3=A5=C3=A0 =C3=A5=C3= =B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B2=C3=A1=C3=A5=C3=B8 =C3= =B2=C3=B8=C3=AA =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=20 %$a+b$=20 %=C3=B0=C3=AE=C3=B6=C3=A0 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=AC=C3=A0 =C3= =A0=C3=B4=C3=B1 =C3=A1=20 %$\Nu(A-(a+b)I)$: %\begin{align*} % (A-(a+b)I|\0)&=3D\bse{cc} -b&b&0\\ a&-a&0 \ese{\widesim{}}\bse{cc} 1&-1&0= \\ 0&0&0 \ese %\end{align*}=09 %=C3=AE=C3=AB=C3=A9=C3=A5=C3=A5=C3=AF =C3=B9=20 %$a^2-b^2=3D1$=20 %=C3=AC=C3=AB=C3=AF=20 %$a\neq 0$=20 %=C3=A0=C3=A5=20 %$b\neq 0$=20 %=C3=A5=C3=A4=C3=B9=C3=A5=C3=B8=C3=A5=C3=BA =C3=AE=C3=B7=C3=A1=C3=A9=C3=AC= =C3=A5=C3=BA. %=C3=A0=C3=A6 =C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=B7=C3=A7=C3=BA =C3=A5"=C3= =B2 %$\v_1=3D\bm1\\1 \em$.\\ % %=C3=B2=C3=AC =C3=AE=C3=B0=C3=BA =C3=AC=C3=AE=C3=B6=C3=A5=C3=A0 =C3=A5=C3= =B7=C3=A8=C3=A5=C3=B8 =C3=B2=C3=B6=C3=AE=C3=A9 =C3=B2=C3=A1=C3=A5=C3=B8 =C3= =B2=C3=B8=C3=AA =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=20 %$0$=20 %=C3=B0=C3=AE=C3=B6=C3=A0 =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8 =C3=AC=C3=A0 =C3= =A0=C3=B4=C3=B1 =C3=A1=20 %${\Nu(A-(0)I)}$: %\begin{align*} % (A-(0)I|\0)=3D(A|\0)&=3D\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*}=09 %=C3=A0=C3=A6 =C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=B7=C3=A7=C3=BA =C3=A5"=C3= =B2 %$\v_2=3D\bm b\\-a \em$ %=C3=A5=C3=A4=C3=A5=C3=A0 =C3=A0=C3=A9=C3=B0=C3=A5 =C3=A5=C3=B7=C3=A8=C3=A5= =C3=B8 =C3=A4=C3=A0=C3=B4=C3=B1 =C3=AB=C3=A9=20 %$a\neq 0$=20 %=C3=A0=C3=A5=20 %$b\neq 0$=20 %=C3=AB=C3=B4=C3=A9 =C3=B9=C3=AB=C3=A1=C3=B8 =C3=B6=C3=A9=C3=A9=C3=B0=C3=A5= .=09=09 \item=20 =C3=A4=C3=B8=C3=A0=C3=A5 =C3=AB=C3=A9 =C3=A9=C3=B9 =C3=AC=20 $C$=20 =C3=A5=20 $C^{-1}$=20 =C3=A0=C3=A5=C3=BA=C3=AD =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3=B6=C3=AE= =C3=A9=C3=A9=C3=AD. =09 % \answer %=09 % =C3=AC=C3=B4=C3=A9 =C3=B1=C3=B2=C3=A9=C3=B3 =C3=A1' =C3=B9=C3=AC =C3=A4= =C3=B9=C3=A0=C3=AC=C3=A4 =C3=A4=C3=B0=C3=A5=C3=AB=C3=A7=C3=A9=C3=BA =C3=A4= =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6=C3=A4=20 % $C$=20 % =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. % =C3=AC=C3=B4=C3=A9 =C3=B9=C3=A0=C3=AC=C3=A4=20 % \ref{eigen-of-inverse} % =C3=B1=C3=B2=C3=A9=C3=B3 =C3=A1' =C3=AC=C3=AB=C3=AC =C3=B2=C3=B8=C3=AA = =C3=B2=C3=B6=C3=AE=C3=A9=20 % $\lambda$=20 % =C3=B9=C3=AC $C$ =C3=A9=C3=B9 =C3=AC % $C^{-1}$=20 % =C3=A9=C3=B9=C3=B0=C3=AD =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3=B6=C3= =AE=C3=A9=C3=A9=C3=AD=20 % $\frac{1}{\lambda}$. % =C3=AC=C3=B4=C3=A9 =C3=B1=C3=B2=C3=A9=C3=B3 =C3=A2' =C3=B9=C3=AC =C3=A4= =C3=B9=C3=A0=C3=AC=C3=A4 =C3=A4=C3=B0=C3=A5=C3=AB=C3=A7=C3=A9=C3=BA =C3=A4= =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD = =C3=B9=C3=AC $C$ =C3=A4=C3=AD=20 % $a+b,\ a-b$.=20 % =C3=B0=C3=A5=C3=A1=C3=B2 =C3=B9=C3=A4=C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3= =A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD =C3=B9=C3=AC=20 % $C^{-1}$=20 % =C3=A4=C3=AD=20 % $\frac{1}{a+b}, \ \frac{1}{a-b}$. % =C3=B0=C3=BA=C3=A5=C3=AF =C3=AB=C3=A9=20 % $1=3Da^2-b^2=3D(a+b)(a-b)$ % =C3=AC=C3=AB=C3=AF % $\frac{1}{a+b}=3Da-b$ % =C3=A5=20 % $\frac{1}{a-b}=3Da+b$. % =C3=AC=C3=AB=C3=AF =C3=AC=20 % $C^{-1}$=20 %=C3=A9=C3=B9 =C3=A0=C3=A5=C3=BA=C3=AD =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3= =B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD=20 % $a+b, \ a-b$=20 % =C3=AB=C3=AE=C3=A5=20 % $C$. %=09 \item=20 =C3=BA=C3=B0=C3=A5 =C3=A3=C3=A5=C3=A2=C3=AE=C3=A4 =C3=AC=C3=AE=C3=A8=C3=B8= =C3=A9=C3=B6=C3=A4 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4 $Y$ =C3=AE=C3=B1=C3=A3=C3= =B8=20 $2\times 2$=20 =C3=AB=C3=AA =C3=B9=C3=A4=C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3=B6=C3=AE= =C3=A9=C3=A9=C3=AD =C3=B9=C3=AC=20 $Y$=20 =C3=B9=C3=A5=C3=B0=C3=A9=C3=AD =C3=AE=C3=A4=C3=B2=C3=B8=C3=AB=C3=A9=C3=AD = =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD =C3=B9=C3=AC=20 $Y^{-1}$. =09 =09 % \answer %=09 %=C3=B0=C3=B7=C3=A7 =C3=AC=C3=AE=C3=B9=C3=AC %$Y=3D\bm 2&0\\ 0&3 \em$ %=C3=A5=C3=A0=C3=A6=20 %$Y^{-1}=3D\bm \frac{1}{2}&0\\ 0&\frac{1}{3} \em$. %=C3=A4=C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9= =C3=AD =C3=B9=C3=AC=20 %$Y$=20 %=C3=A4=C3=AD=20 %$2, 3$ %)=C3=B0=C3=A9=C3=BA=C3=AF =C3=AC=C3=B7=C3=A7=C3=BA =C3=AC=C3=AE=C3=B9=C3= =AC =C3=A5=C3=B7=C3=A8=C3=A5=C3=B8=C3=A9=C3=AD =C3=B2=C3=B6=C3=AE=C3=A9=C3= =A9=C3=AD=20 %$\bm 1\\ 0\em, \bm 0\\ 1\em$.( %=C3=A1=C3=A0=C3=A5=C3=B4=C3=AF =C3=A3=C3=A5=C3=AE=C3=A4, =C3=B7=C3=AC =C3= =AC=C3=B8=C3=A0=C3=A5=C3=BA =C3=B9=C3=A4=C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3= =A4=C3=B2=C3=B6=C3=AE=C3=A9=C3=A9=C3=AD =C3=B9=C3=AC=20 %$Y^{-1}$=20 %=C3=A4=C3=AD=20 %$\frac{1}{2}, \frac{1}{3}$.=20 %=C3=AC=C3=AB=C3=AF =C3=AC=20 %$Y, Y^{-1}$ %=C3=A9=C3=B9 =C3=B2=C3=B8=C3=AB=C3=A9=C3=AD =C3=B2=C3=B6=C3=AE=C3=A9=C3=A9= =C3=AD =C3=B9=C3=A5=C3=B0=C3=A9=C3=AD. =09 \end{enumerate} \item=20 =C3=B0=C3=A2=C3=A3=C3=A9=C3=B8 =C3=B7=C3=A1=C3=A5=C3=B6=C3=BA =C3=AE=C3=A8= =C3=B8=C3=A9=C3=B6=C3=A5=C3=BA =C3=B2=C3=AC =C3=A9=C3=A3=C3=A9=20 \[\S=3D\left\{ \bm a&-b\\ b&a \em\mid a,b\in \setR, a^2+b^2=3D1 \right\}.\] \begin{enumerate} \item=20 =C3=A4=C3=B8=C3=A0=C3=A5 =C3=B9=C3=AB=C3=AC =C3=AE=C3=A8=C3=B8=C3=A9=C3=B6= =C3=A4 =C3=A1=20 $\S$=20 =C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. =09 % \answer=20 %=09 %=C3=A0=C3=AD=20 %$A=3D\bm a&-b\\ b&a\em\in \S$=20 %=C3=A0=C3=A6=20 %$\det A=3Da^2-(-b^2)=3Da^2+b^2=3D1\neq 0$=20 %=C3=AC=C3=AB=C3=AF=20 %$A$=20 %=C3=A4=C3=B4=C3=A9=C3=AB=C3=A4. \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=AC=C3=AB=C3=AC=20 $A\in \S$=20 =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD=20 $A^{-1}\in \S$. =09 %\answer=09 %=09 %=C3=A0=C3=AD=20 %$A=3D\bm a&-b\\ b&a\em\in \S$=20=09 % =C3=A0=C3=A6 =C3=AC=C3=B4=C3=A9 =C3=A8=C3=B2=C3=B0=C3=A4 =C3=AE=C3=BA=C3= =B7=C3=A9=C3=A9=C3=AD=20 %\[A^{-1}=3D\frac{1}{\det A}\bm a&b\\ -b&a \em=3D\frac{1}{a^2+b^2}\bm a&b\\= -b&a \em=3D\frac{1}{1}\bm a&b\\ -b&a \em=3D\bm a&-(-b)\\ b&a \em\in \S\] %)=C3=A4=C3=BA=C3=B0=C3=A0=C3=A9=20 %$a^2+b^2=3D1$ %=C3=B2=C3=A3=C3=A9=C3=A9=C3=AF =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD.(=09 =09 \item=20 =C3=A4=C3=A5=C3=AB=C3=A9=C3=A7=C3=A5 =C3=B9=C3=AC=C3=AB=C3=AC=20 $A, B\in \S$ =C3=AE=C3=BA=C3=B7=C3=A9=C3=A9=C3=AD=20 $AB\in \S$. =09 %\answer % %=C3=BA=C3=A9=C3=A9=C3=B0=C3=A4=20 % $A=3D % \bm a&-b\\ b&a \em, B=3D % \bm c&-d\\ d&c \em. % $ %=09 %=09 % =C3=AB=C3=B2=C3=BA,=20 % \begin{align*} % AB&=3D\bm a&-b\\ b&a \em % \bm c&-d\\ d&c \em=3D\bm ac-bd&-ad-bc \\bc+ad&ac-bd \em=09 % \end{align*} =09 \item=20 =C3=AE=C3=B6=C3=A0=C3=A5 =C3=AB=C3=AC =C3=A4=C3=B4=C3=BA=C3=B8=C3=A5=C3=B0= =C3=A5=C3=BA =C3=AC=C3=AE=C3=B9=C3=A5=C3=A5=C3=A0=C3=A4=20 $A^2=3D\bm 1&0\\ 0&1 \em$ =C3=AB=C3=A0=C3=B9=C3=B8=20 $A\in \S$. =09 %\answer=20 % %=C3=B0=C3=B7=C3=A7=20 %$a=3D-1, b=3D0$=20 %=C3=A5=C3=B0=C3=B7=C3=A1=C3=AC=20=09 % $A=3D\bm a&-b\\b&a \em=3D\bm -1&0\\0&-1 \em$. % =C3=AE=C3=AB=C3=A0=C3=AF =C3=B0=C3=B7=C3=A1=C3=AC=20 % \[A^2=3D\bm -1&0\\0&-1 \em^2=3D\bm -1&0\\0&-1 \em\bm -1&0\\0&-1 \em=3D\bm= 1&0\\0&1 \em\] %=09 \end{enumerate} %\item=20 % %\begin{enumerate} % \item=20 % \item=20 %\end{enumerate} %\item=20 % %\begin{enumerate} % \item=20 % \item=20 %\end{enumerate} \end{enumerate} \textbf{=C3=A4=C3=A5=C3=B8=C3=A0=C3=A5=C3=BA:} \begin{itemize} \item =C3=AE=C3=A8=C3=AC=C3=A4 =C3=A6=C3=A5 =C3=A0=C3=A9=C3=B0=C3=A4 =C3=AC=C3=A4= =C3=A2=C3=B9=C3=A4. \item=20 =C3=B4=C3=BA=C3=B8=C3=A5=C3=B0=C3=A5=C3=BA =C3=A9=C3=B4=C3=A5=C3=B8=C3=B1= =C3=AE=C3=A5 =C3=AC=C3=AB=C3=AC =C3=A4=C3=AE=C3=A0=C3=A5=C3=A7=C3=B8 =C3=A1= 11 =C3=A1=C3=A9=C3=A5=C3=AC=C3=A9. \end{itemize} \end{document} --=-=-=--