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\raggedright
\raggedbottom
\everymath={\displaystyle}
\def\answerspace{\quad}
\def\answerskip{\medskip}
\def\problemskip{\vskip0.75truein}

% Macros

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\def\get@ns#1{\ifx#1\end\let\next=\relax\set@ns\else
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% Preface

%\magnification=\magstep1
\hsize=7.truein
\hoffset=-0.25truein
\vsize=9.5truein
\voffset=-0.25truein
\parindent=0pt

\def\rank{{\rm rank}}
\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\emph#1{{\it #1}}
\def\frac#1#2{{{#1}\over{#2}}}
\def\b{{\bf b}}
\def\x{{\bf x}}
\def\v{{\bf v}}

\def\problemskip{\vskip0.5in}

% Problems

\vbox{
\noindent
{\bf MATH 228: Intro to Lin Alg and Diff Eqns}\hfill
{\bf Practice Final Exam}\quad{\sl Spring 2003}

\problemskip

%1

\noindent\llap{1.\ }%
Let $A = \bmatrix{2&1\cr1&0\cr2&3}$ and $\b=\bmatrix{26\cr-13\cr0}$.
Determine the least squares solution of the linear system $A\x=\b$.

\beginanswers
{(a)~%
$\bmatrix{7\cr-3}$
\answerspace}
{(b)~%
$\bmatrix{11\cr7\cr5}$
\answerspace}
{(c)~%
$\bmatrix{3\cr12\cr10}$
\answerspace}
{(d)~%
$\bmatrix{2\cr7}$
\answerspace}
{(e)~%
$\bmatrix{5\cr-11}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%2

\noindent\llap{2.\ }%
Let $A = \bmatrix{1&2&3\cr3&2&1}$ and $B =
\bmatrix{1&\hfill1\cr0&-1\cr1&\hfill0}$.  Find $A\cdot B$.

\beginanswers
{(a)~%
$\bmatrix{4&-1\cr4&\hfill1}$
\answerspace}
{(b)~%
$\bmatrix{1&2\cr3&0}$
\answerspace}
{(c)~%
$\bmatrix{2&2&4\cr4&1&1}$
\answerspace}
{(d)~%
$\bmatrix{\hfill4&\hfill4&\hfill4\cr-3&-2&-1\cr\hfill1&\hfill2&\hfill3}$
\answerspace}
{(e)~%
$\bmatrix{4&2\cr2&1\cr2&3}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%3

\noindent\llap{3.\ }%
Let $A = \bmatrix{3&2&0\cr1&1&0\cr0&0&1}$. Find $A^{-1}$.

\beginanswers
{(a)~%
$\bmatrix{1&-2&0\cr-1&3&0\cr0&0&1}$
\answerspace}
{(b)~%
$\bmatrix{1/3&1/2&0\cr1&1&0\cr0&0&1}$
\answerspace}
{(c)~%
$\bmatrix{1&1&0\cr3&-2&0\cr0&0&1}$
\answerspace}
{(d)~%
$\bmatrix{1&0&0\cr0&1/3&-1/2\cr0&-1&1}$
\answerspace}
{(e)~%
$\bmatrix{1/3&0&0\cr0&1/2&0\cr0&0&1}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%4

\noindent\llap{4.\ }%
Compute the determinant of $\bmatrix{4&0&3\cr2&2&-1\cr5&0&6}$.

\beginanswers
{(a)~%
18
\answerspace}
{(b)~%
48
\answerspace}
{(c)~%
8
\answerspace}
{(d)~%
$-30$
\answerspace}
{(e)~%
60
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%5

\noindent\llap{5.\ }%
Let $A = \bmatrix{6&3&2\cr3&2&1\cr2&1&0}$ and $B = \bmatrix{1\cr3\cr6}$.
If $X = \bmatrix{x_1\cr x_2\cr x_3}$ is a solution of $A\cdot X = B$,
determine which of the following gives~$x_1$.

\beginanswers
{(a)~%
$x_1 = \frac{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}
{\det\bmatrix{6&3&2\cr3&2&1\cr2&1&0}}$
\answerspace}
{(b)~%
$x_1 = \frac{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}
{\det\bmatrix{1&6&3\cr3&3&2\cr6&1&0}}$
\answerspace}
{(c)~%
$x_1 =\frac{\det\bmatrix{6&3&2\cr3&2&1\cr2&1&0}}
{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}$
\answerspace}
{(d)~%
$x_1 =\frac{\det\bmatrix{1&6&3\cr3&3&2\cr6&2&1}}
{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}$
\answerspace}
{(e)~%
$x_1 = \frac{\det\bmatrix{6&3&1\cr3&2&3\cr2&1&6}}
{\det\bmatrix{6&3&1\cr3&2&3\cr2&1&3}}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%6

\noindent\llap{6.\ }%
Determine which of the following matrices gives $R$ in the
$QR$~decomposition of
$\bmatrix{1 & 0 & 0 \cr 0 & 1 & 1 \cr 1 & 0 & 1 \cr 0 & 1 & 0}$.

\beginanswers
{(a)~%
$\bmatrix{\sqrt{2} & 0 & 1/\sqrt{2} \cr
  0 &\sqrt{2}  & 1/\sqrt{2} \cr 0 & 0 & 1}$
\answerspace}
{(b)~%
$\bmatrix{\sqrt{2} & 0 & 1/\sqrt{3} \cr
  0 &\sqrt{2}  & 2/\sqrt{3} \cr 0 & 0 & 1}$
\answerspace}
{(c)~%
$\bmatrix{\sqrt{2} & 0 & -1/\sqrt{6} \cr
  0 &\sqrt{2}  & 1/\sqrt{6} \cr 0 & 0 & 1}$
\answerspace}
{(d)~%
$\bmatrix{\sqrt{2} & 0 & 1/\sqrt{3} \cr
  0 &\sqrt{2}  & 1/\sqrt{3} \cr 0 & 0 & 1}$
\answerspace}
{(e)~%
$\bmatrix{\sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} \cr
  0 &\sqrt{2}  & 1/\sqrt{2} \cr 0 & 0 & 1/\sqrt{2}}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%7

\noindent\llap{7.\ }%
Determine which of the following statements is \emph{not} equivalent to
$A$ being an $n\times n$ orthogonal matrix.

\beginanswers
{(a)~%
$A \v$ is orthogonal to $\v$ for all vectors $\v$ in
$R^{n}$.
\answerspace}
{(b)~%
The column vectors of $A$ are orthonormal.
\answerspace}
{(c)~%
The row vectors of $A$ are orthonormal.
\answerspace}
{(d)~%
$A^{-1} = A^{T}$
\answerspace}
{(e)~%
$\|A \v \| = \| \v \|$ for all vectors $\v$ in $R^{n}$.
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%8

\noindent\llap{8.\ }%
A matrix $A$ has eigenvectors $\bmatrix{1 \cr 0 \cr 0}$, $\bmatrix{1 \cr
1 \cr 0}$, and $\bmatrix{1 \cr 1 \cr 1}$ with corresponding eigenvalues
$2$, $-1$, and $-2$. Use this information to reconstruct the matrix~$A$.

\beginanswers
{(a)~%
$\bmatrix{2 & -3 & -1 \cr 0 & -1 & -1 \cr 0 & 0 & -2}$
\answerspace}
{(b)~%
$\bmatrix{2 & 0 & 0 \cr 0 & -1 & 0 \cr 0 & 0 & -2}$
\answerspace}
{(c)~%
$\bmatrix{1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1}$
\answerspace}
{(d)~%
$\bmatrix{2 & -1 & -2 \cr 0 & -1 & -2 \cr 0 & 0 & -2}$
\answerspace}
{(e)~%
$\bmatrix{ 2 & -3 & 1 \cr 0 & -1 & 1 \cr 0 & 0 & -2}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%9

\noindent\llap{9.\ }%
Find a basis for the null space of $\bmatrix{\hfill1&\hfill2&-1&\hfill0\cr
-3&-4&\hfill1&-4\cr\hfill2&\hfill5&-3&\hfill-2}$.

\beginanswers
{(a)~%
$\left\{\bmatrix{-1\cr\hfill1\cr\hfill1\cr\hfill0},
  \bmatrix{-4\cr\hfill2\cr\hfill0\cr\hfill1}\right\}$
\answerspace}
{(b)~%
$\left\{\bmatrix{\hfill1\cr\hfill2\cr-1\cr\hfill0},
  \bmatrix{\hfill0\cr\hfill1\cr-1\cr-2}\right\}$
\answerspace}
{(c)~%
$\left\{\bmatrix{1\cr0\cr0},\bmatrix{2\cr1\cr0},\bmatrix{-1\cr-1\cr0},
  \bmatrix{\hfill0\cr-2\cr\hfill0}\right\}$
\answerspace}
{(d)~%
$\left\{\bmatrix{\hfill1\cr-3\cr\hfill2},
  \bmatrix{\hfill2\cr-4\cr\hfill5}\right\}$
\answerspace}
{(e)~%
$\left\{\bmatrix{1\cr0\cr0\cr0},\bmatrix{0\cr1\cr0\cr0},
  \bmatrix{0\cr0\cr0\cr1} \right\}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%10

\noindent\llap{10.\ }%
Let $W = {\rm Span}\{1+x+x^2,2+3x+3x^2+x^3,1-x^3,x+x^2+x^3\}$.
Find the dimension of $W$.

\beginanswers
{(a)~%
2
\answerspace}
{(b)~%
1
\answerspace}
{(c)~%
3
\answerspace}
{(d)~%
4
\answerspace}
{(e)~%
5
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%11

\noindent\llap{11.\ }%
Let $B=\left\{\bmatrix{1&0\cr1&0}, \bmatrix{0&1\cr0&1}, \bmatrix{0&0\cr1&0},
\bmatrix{0&0\cr0&1}\right\}$ be a basis for the vector space of $2\times 2$
matrices and let $\v = \bmatrix{2&-1\cr5&3}$. Find $[\v]_B$, the coordinates
of $\v$ with respect to the basis~$B$.

\beginanswers
{(a)~%
$\bmatrix{\hfill2\cr-1\cr\hfill3\cr\hfill4}$
\answerspace}
{(b)~%
$\bmatrix{\hfill4\cr\hfill3\cr-1\cr\hfill2}$
\answerspace}
{(c)~%
$\bmatrix{\hfill2\cr-1\cr\hfill5\cr\hfill3}$
\answerspace}
{(d)~%
$\bmatrix{\hfill2\cr\hfill5\cr-1\cr\hfill3}$
\answerspace}
{(e)~%
$\bmatrix{1\cr0\cr1\cr0}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%12

\noindent\llap{12.\ }%
Let $A$ be a $3\times 3$ matrix whose null space is spanned by
$\bmatrix{1\cr0\cr0}$. Determine which of the following statements
is true.

\beginanswers
{(a)~%
$\rank A = 2$
\answerspace}
{(b)~%
$\det A \ne 0$
\answerspace}
{(c)~%
$A$ is invertible.
\answerspace}
{(d)~%
$\rank A = 1$
\answerspace}
{(e)~%
$A$ must be symmetric.
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%13

\noindent\llap{13.\ }%
Suppose that $y_1$ and $y_2$ are solutions of $ty'' - y' + e^{t^2}y = 0$ and
their Wronskian satisfies\hfill\break
$W(y_1,y_2)(1) = 1$. Find $W(y_1,y_2)(t)$.

\beginanswers
{(a)~%
$t$
\answerspace}
{(b)~%
$e^{t^2}$
\answerspace}
{(c)~%
$1/t$
\answerspace}
{(d)~%
$e^{1-t}$
\answerspace}
{(e)~%
\emph{cannot be determined}
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%14

\noindent\llap{14.\ }%
Solve the initial value problem $2y'' - 12y' + 18y = 0$,\quad $y(0) = 1$,\quad
$y'(0) = 2$.

\beginanswers
{(a)~%
$y = (1-t)e^{3t}$
\answerspace}
{(b)~%
$y = e^{3t}$
\answerspace}
{(c)~%
$y = e^{3t}-t$
\answerspace}
{(d)~%
$y = e^{-3t} + 5t e^{-3t}$
\answerspace}
{(e)~%
$y = e^t\cos(3t) + e^t \sin(3t)$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%15

\noindent\llap{15.\ }%
Find the general solution of $y'' - 4y' + 13y = 0$.

\beginanswers
{(a)~%
$c_1 e^{2t}\cos(3t) + c_2 e^{2t}\sin(3t)$
\answerspace}
{(b)~%
$c_1 e^{3t}\cos(2t) + c_2 e^{3t}\sin(2t)$
\answerspace}
{(c)~%
$c_1 \cos(2t) + c_2 \sin(2t)$
\answerspace}
{(d)~%
$c_1 e^{2t}+ c_2 e^{-2t}$
\answerspace}
{(e)~%
$c_1 e^{5t}+ c_2 e^{-t}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%16

\noindent\llap{16.\ }%
Find the solution to the initial value problem
$y^2 y' + x y^3 + x = 0$,\quad $y(2)=1$.

\beginanswers
{(a)~%
$\big( 2 e^{6-3x^{2}/2} - 1 \big)^{1/3}$
\answerspace}
{(b)~%
$e^{-2 + x^{2}/2}$
\answerspace}
{(c)~%
$\big( 2 e^{6-3x^{2}/2} \big)^{1/3} - 1 $
\answerspace}
{(d)~%
$\big( e^{-3x^{2}/2} - 1 \big)^{1/3}$
\answerspace}
{(e)~%
$e^{2 - x^{2}/2}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%17

\noindent\llap{17.\ }%
Determine which of the following functions is an integrating factor for
the equation\hfil\break $x \, dx + (x^{2}+ e^{y^{2}}) \, dy = 0$.

\beginanswers
{(a)~%
$e^{2y}$
\answerspace}
{(b)~%
$2y$
\answerspace}
{(c)~%
$e^{-2x}$
\answerspace}
{(d)~%
$\frac{1}{x}$
\answerspace}
{(e)~%
$x e^{2y}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%18

\noindent\llap{18.\ }%
A tank initially contains 100 liters of pure water. A mixture containing
a concentration of 2 grams per liter of salt enters the tank at a rate of
3 liters per minute, and the well-stirred mixture leaves the tank at
the same rate.  Find the amount of salt, in grams, in the tank after
$t$ minutes.

\beginanswers
{(a)~%
$200 - 200 e^{-0.03t}$
\answerspace}
{(b)~%
$e^{6t} -1$
\answerspace}
{(c)~%
$6t$
\answerspace}
{(d)~%
$100 e^{-6t} - 100$
\answerspace}
{(e)~%
$200 e^{0.03t} -200$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%19

\noindent\llap{19.\ }%
Find the form of a particular solution of $y'' + 2y' + y = e^{-t}(\sin(t)
+ t)$.

\beginanswers
{(a)~%
$y_p = e^{-t}\big(A\sin(t) + B\cos(t) + C t^2 + D t^3\big)$
\answerspace}
{(b)~%
$y_p = t^2e^{-t}\big(A\sin(t) + B\cos(t) + C + D t\big)$
\answerspace}
{(c)~%
$y_p = e^{-t}\big(A\sin(t) + B\cos(t) + C t^3\big)$
\answerspace}
{(d)~%
$y_p = t^2e^{-t}\big(A\sin(t) + B\cos(t) + C\big)$
\answerspace}
{(e)~%
$y_p = e^{-t}\big(A\sin(t) + B\cos(t) + C + D t\big)$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%20

\noindent\llap{20.\ }%
Given that $y_1 = e^t$ and $y_2 = t$ are solutions of
$(1-t)y'' + ty' - y = 0$, find a particular solution of
$(1-t)y'' + ty' - y = e^t(1-t)^2$.

\beginanswers
{(a)~%
$y_p = e^t\big(t - t^2/2\big)$
\answerspace}
{(b)~%
$y_p = e^t(1 - t)$
\answerspace}
{(c)~%
$y_p = e^{2t} + te^t$
\answerspace}
{(d)~%
$y_p = t e^t + t^2$
\answerspace}
{(e)~%
$y_p = e^t(1-t)^3/3$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

% Appendix

{}

\end
