<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%=========================================================================%
% initialize TeX
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\magnification=\magstep1
\baselineskip=14pt
\hsize = 6.5truein
\hoffset = 0.0truein
\vsize = 9.1truein
\voffset = -0.25truein
\emergencystretch = 0.05\hsize

\newif\ifblackboardbold

% comment out the following line if AMS msbm fonts aren't available
\blackboardboldtrue

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% select fonts
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\font\titlefont=cmbx12 scaled\magstephalf
\font\sectionfont=cmbx12
\font\netfont=cmtt9

% Establish AMS blackboard bold fonts without using amssym.def, amssym.tex

\newfam\bboldfam
\ifblackboardbold
\font\tenbbold=msbm10
\font\sevenbbold=msbm7
\font\fivebbold=msbm5
\textfont\bboldfam=\tenbbold
\scriptfont\bboldfam=\sevenbbold
\scriptscriptfont\bboldfam=\fivebbold
\def\bbold{\fam\bboldfam\tenbbold}
\else
\def\bbold{\bf}
\fi

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% widow control
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% usage:
% \widow{.2} % start new page if &lt;.2 page left

\def\widow#1{\vskip 0pt plus#1\vsize\goodbreak\vskip 0pt plus-#1\vsize}

%=========================================================================%
% exam macros (use \practice or \exam to initialize)
%=========================================================================%

\newcount\probcount

\def\practice{
\probcount=0
\def\prob##1\par
{\medskip\global\advance\probcount by 1\noindent
{\bf[\number\probcount]}~##1}}

\def\exam{
\probcount=0
\def\prob##1\par
{\global\advance\probcount by 1\noindent {\bf[\number\probcount]}~##1
\vfill\eject
\noindent{\bf Problem: \hbox to 30pt{\hrulefill}}
\vfill\eject
}
\footline={\tenbf Page \folio\hfil Continued on page: \hbox to
30pt{\hrulefill}}
% set up name, id, school fields for exam
\vskip 0.1in 
{\bf \noindent Name: \hrulefill \hbox{}}
\vskip 0.1in 
{\bf \noindent ID: \hrulefill ~ School: \hrulefill \hbox{}}
\vskip 0.2in
}

% set up those boxes to put the scores in

\def\boxAsub#1{\vbox{\hbox{#1}}}
\def\boxA#1#2{[{\bf #1}] (#2 pts)}
\def\boxT{\bf TOTAL}
\def\boxB{\vbox to 25pt{\hbox{\kern 56pt}}}

\def\fiveboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &amp;&amp;\hfil##\hfil\vrule\cr
 \noalign{\hrule}
 \boxA 1{#1} &amp;
 \boxA 2{#1} &amp;
 \boxA 3{#1} &amp;
 \boxA 4{#1} &amp;
 \boxA 5{#1} &amp;
 \boxT \cr
 \noalign{\hrule}
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB  \cr
 \noalign{\hrule}
}}}}

\def\eightboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &amp;&amp;\hfil##\hfil\vrule\cr
 \multispan4{\hrulefill}\cr
 \boxA 1{#1} &amp;
 \boxA 2{#1} &amp;
 \boxA 3{#1} &amp;
 \boxA 4{#1} \cr
 \multispan4{\hrulefill}\cr
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB \cr
 \noalign{\hrule}
 \boxA 5{#1} &amp;
 \boxA 6{#1} &amp;
 \boxA 7{#1} &amp;
 \boxA 8{#1} &amp;
 \boxT \cr
 \noalign{\hrule}
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB\cr
 \noalign{\hrule}
}}}}

\def\tenboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &amp;&amp;\hfil##\hfil\vrule\cr
 \multispan5{\hrulefill}\cr
 \boxA 1{#1} &amp;
 \boxA 2{#1} &amp;
 \boxA 3{#1} &amp;
 \boxA 4{#1} &amp;
 \boxA 5{#1} \cr
 \multispan5{\hrulefill}\cr
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB \cr
 \noalign{\hrule}
 \boxA 6{#1} &amp;
 \boxA 7{#1} &amp;
 \boxA 8{#1} &amp;
 \boxA 9{#1} &amp;
 \boxA{10}{#1} &amp;
 \boxT \cr
 \noalign{\hrule}
 \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB &amp; \boxB\cr
 \noalign{\hrule}
}}}}

%==========================================================================
% macros
%==========================================================================

\def\bmatrix#1{{\left[\vcenter{\halign
  {&amp;\kern 4pt\hfil$##\mathstrut$\kern 4pt\cr#1}}\right]}}

% surround with $ $ if not already in math mode
\def\enma#1{{\ifmmode#1\else$#1$\fi}}

\def\mathbb#1{{\bbold #1}}
\def\mathbf#1{{\bf #1}}

% blackboard bold symbols
\def\FF{\enma{\mathbb{F}}}
\def\NN{\enma{\mathbb{N}}}
\def\RR{\enma{\mathbb{R}}}
\def\CC{\enma{\mathbb{C}}}
\def\QQ{\enma{\mathbb{Q}}}
\def\ZZ{\enma{\mathbb{Z}}}
\def\bbPP{\enma{\mathbb{P}}}

% caligraphic symbols
\def\cAA{\enma{\cal A}}
\def\cBB{\enma{\cal B}}

% bold symbols
\def\aa{\enma{\mathbf{a}}}
\def\bb{\enma{\mathbf{b}}}
\def\cc{\enma{\mathbf{c}}}
\def\ee{\enma{\mathbf{e}}}
\def\mm{\enma{\mathbf{m}}}
\def\pp{\enma{\mathbf{p}}}
\def\vv{\enma{\mathbf{v}}}
\def\ww{\enma{\mathbf{w}}}
\def\uu{\enma{\mathbf{u}}}
\def\vv{\enma{\mathbf{v}}}
\def\xx{\enma{\mathbf{x}}}
\def\yy{\enma{\mathbf{y}}}
\def\XX{\enma{\mathbf{X}}}
\def\KK{\enma{\mathbf{K}}}
\def\PP{\enma{\mathbf{P}}}
\def\LL{\enma{\mathbf{L}}}
\def\II{\enma{\mathbf{I}}}
\def\zero{\enma{\mathbf{0}}}
\def\blambda{\enma{\mathbf{\lambda}}}

% misc

\def\abs#1{\enma{\left| #1 \right|}}
\def\set#1{\enma{\{#1\}}}
\def\setdef#1#2{\enma{\{\;#1\;\,|\allowbreak
  \;\,#2\;\}}}
\def\idealdef#1#2{\enma{\langle\;#1\;\,
  |\;\,#2\;\rangle}}

%=========================================================================%
% exam
%=========================================================================%

\centerline{\titlefont Final Exam}
\smallskip
\centerline{Linear Algebra, Dave Bayer, December 16, 1999}
\medskip

%\practice
\exam
\eightboxes{5}

Please work only one problem per page, starting with the pages provided, and
number all continuations clearly. Only work which can be found in this way will
be graded.

Please do not use calculators or decimal notation.

\bigskip

\prob
Let $L$ be the linear transformation from $\RR^3$ to $\RR^3$ which projects
onto the line $(1,1,1)$. In other words, if $\uu$ is a vector in
$\RR^3$, then $L(\uu)$ is the projection of $\uu$ onto the vector $(1,1,1)$.
Choose a basis $\set{\vv_1, \vv_2, \vv_3}$ for $\RR^3$, and find a matrix $A$
representing $L$ with respect to this basis.

\prob
Compute the determinant of the following $4\times 4$ matrix:
$$\bmatrix{
2 &amp; -1 &amp; 0 &amp; 0 \cr
-1 &amp; 2 &amp; -1 &amp; 0 \cr
0 &amp; -1 &amp; 2 &amp; -1 \cr
0 &amp; 0 &amp; -1 &amp; 2 \cr
}$$
What can you say about the determinant of the $n\times n$ matrix with the same
pattern?



\prob
By least squares, find the equation of the form $y=ax+b$ which best fits the
data  $(x_1,y_1)=(0,0)$, $(x_2,y_2)=(1,1)$, $(x_3,y_3)=(3,1)$.

\prob
Find $(s,t)$ so $\bmatrix{1 &amp; 0\cr 0 &amp; 1\cr 1 &amp; 1\cr }
\bmatrix{s\cr t\cr}$ is as close as possible to $\bmatrix{1\cr 1\cr 0\cr}$.

\prob
Find an orthogonal basis for the subspace $w+2x+3y+4z=0$ of $\RR^4$.

\prob
Let $A$ be the matrix
$$A \;=\;
\bmatrix{3 &amp; 2\cr 2 &amp; 3\cr }.$$
Find a basis of eigenvectors and eigenvalues for $A$.
Find the matrix exponential $e^A$.

\prob
Let $A$ be the matrix
$$A \;=\;
\bmatrix{2 &amp; 0 &amp; 1\cr 0 &amp; 3 &amp; 0\cr 1 &amp; 0 &amp; 2\cr }.$$
Find an orthogonal basis in which $A$ is diagonal.

\prob
Find a matrix $A$ so the substitution
$$\bmatrix{x \cr y \cr} = A \bmatrix{s \cr t \cr}$$
transforms the quadratic form
$x^2+4xy+y^2$ into the quadratic form $s^2-t^2$.

\bye</pre></body></html>