%=========================================================================%
% initialize TeX
%=========================================================================%

\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

%=========================================================================%
% select fonts
%=========================================================================%

\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

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

\newcount\probcount
\probcount=0

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

\def\exam{
\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
 &&\hfil##\hfil\vrule\cr
 \noalign{\hrule}
 \boxA 1{#1} &
 \boxA 2{#1} &
 \boxA 3{#1} &
 \boxA 4{#1} &
 \boxA 5{#1} &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB & \boxB  \cr
 \noalign{\hrule}
}}}}

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

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

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

\def\bmatrix#1{{\left[\vcenter{\halign
  {&\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\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 Practive Exam 1}
\smallskip
\centerline{Linear Algebra, Dave Bayer, September 30, 1999}
\medskip

%\practice
\exam
\fiveboxes{6}

To be graded, this practice exam must be
turned in at the end of class on Thursday, September 30. Such exams will be
returned  in class on the following Tuesday, October 5. Participation is
optional; scores will not be used to determine course grades. If you do
participate, you may use your judgement in seeking any assistance of your
choosing, or you may take this test under simulated exam conditions.  If you
don't participate, you are electing to join the control group.

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
Solve the following system of equations:
$$
\bmatrix
{2 & -1 & 0 & 0 \cr
-1 & 2 & -1 & 0 \cr
0 & -1 & 2 & -1 \cr
0 & 0 & -1 & 2 \cr}
\bmatrix{w \cr x \cr y \cr z \cr}
=
\bmatrix{1 \cr 0 \cr 0 \cr 6 \cr}
$$

\prob
Compute a matrix giving the number of walks of length 4
between pairs of vertices of the following graph:
\input epsf
$$\epsfbox{graph.eps}$$

\prob
Express the following matrix as a product of elementary matrices:
$$\bmatrix{
0 & 1 & 3 & 0 \cr
0 & 0 & 1 & 4 \cr
0 & 0 & 0 & 1 \cr
2 & 0 & 0 & 0 \cr
}$$


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


\prob
Use Cramer's rule to give a formula for $w$ in the solution to the following
system of equations:
$$
\bmatrix
{2 & -1 & 0 & 0 \cr
-1 & 2 & -1 & 0 \cr
0 & -1 & 2 & -1 \cr
0 & 0 & -1 & 2 \cr}
\bmatrix{w \cr x \cr y \cr z \cr}
=
\bmatrix{a \cr b \cr c \cr d \cr}
$$

\bye