%=========================================================================%
% 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 Exam 2}
\smallskip
\centerline{Linear Algebra, Dave Bayer, November 11, 1999}
\medskip

%\practice
\exam
\fiveboxes{6}

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 degree $\le 4$ polynomials in one variable $x$ with
real coefficients. Let
$Q$ be the subset of
$P$ consisting of all odd polynomials, i.e. all polynomials $f(x)$ so $f(-x) =
-f(x)$. Show that $Q$ is a subspace of $P$.
Choose a basis
for $Q$. Extend this basis for $Q$ to a basis for $P$.

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

\prob
Let $L$ be the linear transformation from $\RR^3$ to $\RR^3$ which reflects
through the plane $P$ defined by $x+y+z=0$. In other words, if $\uu$ is a vector
lying in the plane $P$, and $\vv$ is a vector perpendicular to the plane
$P$, then $L(\uu+\vv)=\uu-\vv$.
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
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
$$
\ee_1 \;=\; \bmatrix{1 \cr 0 \cr}, \quad
\ee_2 \;=\; \bmatrix{0 \cr 1 \cr}, \quad
\vv_1 \;=\; \bmatrix{1 \cr 1 \cr}, \quad
\vv_2 \;=\; \bmatrix{1 \cr 2 \cr}, \quad
A \;=\; \bmatrix{-1 & 2 \cr -4 & 5\cr}.$$
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 $\set{\vv_1,\;
\vv_2}$.

\prob
Let $\set{\uu_1,\; \uu_2}$, $\set{\vv_1,\; \vv_2}$, and $\set{\ww_1,\;
\ww_2}$ be ordered bases for $\RR^2$. If $$A \;=\; \bmatrix{1 & 2 \cr 0 &
1\cr}$$ is the transition matrix corresponding to the change of basis from
$\set{\uu_1,\; \uu_2}$ to
$\set{\vv_1,\; \vv_2}$, and $$B \;=\; \bmatrix{1 & 0 \cr 3 & 1\cr}$$ is the
transition matrix corresponding to the change of basis from $\set{\uu_1,\;
\uu_2}$ to
$\set{\ww_1,\; \ww_2}$,
express $\vv_1$ and $\vv_2$ in terms of $\ww_1$ and $\ww_2$.



\bye