%=========================================================================% % 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 %=========================================================================% % widow control %=========================================================================% % usage: % \widow{.2} % start new page if <.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 &&\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 Practice Exam 1} \practice \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: $$ \bmatrix {2 & -1 & 0 & 0 \cr -1 & 2 & -1 & 0 \cr 0 & -1 & 2 & -1 \cr 0 & 0 & -1 & 2 \cr} \bmatrix{w \cr x \cr y \cr z \cr} = \bmatrix{a \cr b \cr c \cr d \cr} $$ \vfill\eject \centerline{\titlefont Exam 1} \practice \prob Solve the following system of equations: $$ \bmatrix {0 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & 0 \cr} \bmatrix{x \cr y \cr z \cr} = \bmatrix{2 \cr 0 \cr 0 \cr} $$ \prob Compute matrices giving the number of walks of lengths 1, 2, and 3 between pairs of vertices of the following graph: \input epsf $$\epsfbox{graph2.eps}$$ \prob Express the following matrix as a product of elementary matrices: $$\bmatrix{ 0 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & 0 \cr }$$ \prob Compute the determinant of the following $4\times 4$ matrix: $$\bmatrix{ 1 & 1 & 1 & 0 \cr 2 & 2 & 0 & 2 \cr 3 & 0 & 3 & 3 \cr 0 & 4 & 4 & 4 \cr }$$ 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 the solution to the following system of equations: $$ \bmatrix {0 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & 0 \cr} \bmatrix{x \cr y \cr z \cr} = \bmatrix{2a \cr 2b \cr 2c\cr} $$ \vfill\eject \centerline{\titlefont Practice Exam 2} \practice \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 $\set{\vv_1,\; \vv_2}$. \vfill\eject \centerline{\titlefont Exam 2} \practice \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 $\ww_2$. \vfill\eject \centerline{\titlefont Additional Practice Problems for Final} \practice \prob By least squares, find the equation of the form $y=ax+b$ which best fits the data $(x_1,y_1)=(0,1)$, $(x_2,y_2)=(1,1)$, $(x_3,y_3)=(2,-1)$. \prob Find $(s,t)$ so $\bmatrix{1 & 0\cr -1 & 1\cr 0 & -1\cr } \bmatrix{s\cr t\cr}$ is as close as possible to $\bmatrix{1\cr 0\cr 0\cr}$. \prob Find an orthogonal basis for the subspace $w+x+y+z=0$ of $\RR^4$. \prob Let $A$ be the matrix $$A \;=\; \bmatrix{-3 & -4\cr -4 & 3\cr }.$$ Find a basis of eigenvectors and eigenvalues for $A$. Find the matrix exponential $e^A$. \prob Find a matrix $A$ in standard coordinates having eigenvectors $\vv_1=(1,1)$, $\vv_2=(1,2)$ with corresponding eigenvalues $\lambda_1=2$, $\lambda_2=-1$. \prob Let $A$ be the matrix $$A \;=\; \bmatrix{2 & 1 & 1\cr 1 & 2 & 1\cr 1 & 1 & 2\cr }.$$ Find an orthogonal basis in which $A$ is diagonal. \bye