%=========================================================================%
% 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\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 Practive Exam 2}
\smallskip
\centerline{Linear Algebra, Dave Bayer, October 28, 1999}
\medskip

%\practice
\exam
\fiveboxes{6}

To be graded, this practice exam must be
turned in at the end of class on Thursday, November 4. Such exams will be
returned  in class on the following Tuesday, November 9. 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.

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 $P$ be the set of all polynomials $f(x)$, and let $Q$ be the subset of $P$
consisting of all polynomials $f(x)$ so $f(0) = f(1) = 0$. Show that $Q$ is a
subspace of $P$.

\prob
Let $A$ be the matrix
$$A \;=\;
\bmatrix{1 & -1 & 1\cr 2 & -2 & 2 \cr 1 & -1 & 0 \cr}.$$
Compute the row space and column space of $A$.

\prob
The four vectors
$$
\vv_1 \;=\; \bmatrix{1 \cr 0 \cr 2 \cr}, \quad
\vv_2 \;=\; \bmatrix{-1 \cr 0 \cr -2 \cr}, \quad
\vv_3 \;=\; \bmatrix{1 \cr 2 \cr 6 \cr}, \quad
\vv_4 \;=\; \bmatrix{0 \cr 1 \cr 2 \cr}$$
span a subspace $V$ of $\RR^3$, but are not a basis for $V$.
Choose a subset of $\set{\vv_1,\; \vv_2,\; \vv_3,\; \vv_4}$ which forms a basis
for $V$. Extend this basis for $V$ to a basis for $\RR^3$.


\prob
Let $L$ be the linear transformation from $\RR^3$ to $\RR^3$ which rotates one
half turn around the axis given by the vector $(1,1,1)$. Find a matrix $A$
representing $L$ with respect to the standard basis
$$
\ee_1 \;=\; \bmatrix{1 \cr 0 \cr 0 \cr}, \quad
\ee_2 \;=\; \bmatrix{0 \cr 1 \cr 0 \cr}, \quad
\ee_3 \;=\; \bmatrix{0 \cr 0 \cr 1 \cr}.$$
Choose a new basis $\set{\vv_1,\; \vv_2, \vv_3}$ for $\RR^3$ which makes $L$
easier to describe, and find a matrix $B$
representing $L$ with respect to this new basis.

\prob
Let $\set{\ee_1,\; \ee_2}$ and $\set{\vv_1,\; \vv_2}$ be ordered bases for
$\RR^2$, and let $L$ be the linear transformation represented by the matrix $A$
with respect to $\set{\ee_1,\; \ee_2}$, where
$$
\ee_1 \;=\; \bmatrix{1 \cr 0 \cr}, \quad
\ee_2 \;=\; \bmatrix{0 \cr 1 \cr}, \quad
\vv_1 \;=\; \bmatrix{2 \cr 1 \cr}, \quad
\vv_2 \;=\; \bmatrix{1 \cr -2 \cr}, \quad
A \;=\; \bmatrix{6 & -2 \cr -2 & 9\cr}.$$
Find the transition matrix $S$
corresponding to the change of basis from
$\set{\ee_1,\; \ee_2}$ to $\set{\vv_1,\; \vv_2}$.
Find a matrix $B$ representing $L$ with respect to $\set{\vv_1,\;
\vv_2}$.

\bye