\magnification=\magstep1
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\def\v{{\bf v}}
\def\rank{{\rm rank}}
\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\emph#1{{\it #1}}
\def\frac#1#2{{{#1}\over{#2}}}

\def\problemspace{0.5in}

%% TITLE
\noindent
{\bf MATH 228: Intro to Lin Alg and Diff Eqns}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl May 7, 2002}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}

0.5truein
\noindent
Record your answers by placing an $\times$ through one letter for each
problem on this page. There are 20 questions worth 6 points each. You
start with 30 points.

\bigskip

You may \emph{not} use a calculator.

\bigskip

\def\answerspace{0.25in}


%1
%% PROBLEM
Determine which of the following matrices gives $U$ in an
$LU$~decomposition of
$\bmatrix{1&2&0&4\cr1&0&2&4\cr0&1&2&4}$.

%% CORRECT
$\bmatrix{1&2&0&4\cr0&1&-1&0\cr0&0&3&4}$
%% WRONG
$\bmatrix{1&2&0&4\cr0&0&0&1\cr0&0&0&0}$
%% WRONG
$\bmatrix{1&2&0&4\cr0&0&1&-2\cr0&1&0&0}$
%% WRONG
$\bmatrix{1&2&0&4\cr0&1&0&-2\cr0&0&1&2}$
%% WRONG
$\bmatrix{1&2&0&4\cr0&1&2&4\cr0&0&0&1}$


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

%% CORRECT
$\bmatrix{4&-1\cr4&\hfill1}$
%% WRONG
$\bmatrix{1&2\cr3&0}$
%% WRONG
$\bmatrix{2&2&4\cr4&1&1}$
%% WRONG
$\bmatrix{\hfill4&\hfill4&\hfill4\cr-3&-2&-1\cr\hfill1&\hfill2&\hfill3}$
%% WRONG
$\bmatrix{4&2\cr2&1\cr2&3}$


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

%% CORRECT
$\bmatrix{1&-2&0\cr-1&3&0\cr0&0&1}$
%% WRONG
$\bmatrix{1/3&1/2&0\cr1&1&0\cr0&0&1}$
%% WRONG
$\bmatrix{1&1&0\cr3&-2&0\cr0&0&1}$
%% WRONG
$\bmatrix{1&0&0\cr0&1/3&-1/2\cr0&-1&1}$
%% WRONG
$\bmatrix{1/3&0&0\cr0&1/2&0\cr0&0&1}$


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

%% CORRECT
18
%% WRONG
48
%% WRONG
8
%% WRONG
-30
%% WRONG
60


%5
%% PROBLEM
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~$x_1$.

%% CORRECT
$x_1 = \frac{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}
{\det\bmatrix{6&3&2\cr3&2&1\cr2&1&0}}$
%% WRONG
$x_1 = \frac{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}
{\det\bmatrix{1&6&3\cr3&3&2\cr6&1&0}}$
%% WRONG
$x_1 =\frac{\det\bmatrix{6&3&2\cr3&2&1\cr2&1&0}}
{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}$
%% WRONG
$x_1 =\frac{\det\bmatrix{1&6&3\cr3&3&2\cr6&2&1}}
{\det\bmatrix{1&3&2\cr3&2&1\cr6&1&0}}$
%% WRONG
$x_1 = \frac{\det\bmatrix{6&3&1\cr3&2&3\cr2&1&6}}
{\det\bmatrix{6&3&1\cr3&2&3\cr2&1&3}}$


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

%% CORRECT
$\bmatrix{\sqrt{2} & 0 & 1/\sqrt{2} \cr
  0 &\sqrt{2}  & 1/\sqrt{2} \cr 0 & 0 & 1}$
%% WRONG
$\bmatrix{\sqrt{2} & 0 & 1/\sqrt{3} \cr
  0 &\sqrt{2}  & 2/\sqrt{3} \cr 0 & 0 & 1}$
%% WRONG
$\bmatrix{\sqrt{2} & 0 & -1/\sqrt{6} \cr
  0 &\sqrt{2}  & 1/\sqrt{6} \cr 0 & 0 & 1}$
%% WRONG
$\bmatrix{\sqrt{2} & 0 & 1/\sqrt{3} \cr
  0 &\sqrt{2}  & 1/\sqrt{3} \cr 0 & 0 & 1}$
%% WRONG
$\bmatrix{\sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} \cr
  0 &\sqrt{2}  & 1/\sqrt{2} \cr 0 & 0 & 1/\sqrt{2}}$

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

%% CORRECT
$A \v$ is orthogonal to $\v$ for all vectors $\v$ in
$R^{n}$.
%% WRONG
The column vectors of $A$ are orthonormal.
%% WRONG
The row vectors of $A$ are orthonormal.
%% WRONG
$A^{-1} = A^{T}$
%% WRONG
$\|A \v \| = \| \v \|$ for all vectors $\v$ in $R^{n}$.

%8
%% PROBLEM
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~$A$.

%% CORRECT
$\bmatrix{2 & -3 & -1 \cr 0 & -1 & -1 \cr 0 & 0 & -2}$
%% WRONG
$\bmatrix{2 & 0 & 0 \cr 0 & -1 & 0 \cr 0 & 0 & -2}$
%% WRONG
$\bmatrix{1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1}$
%% WRONG
$\bmatrix{2 & -1 & -2 \cr 0 & -1 & -2 \cr 0 & 0 & -2}$
%% WRONG
$\bmatrix{ 2 & -3 & 1 \cr 0 & -1 & 1 \cr 0 & 0 & -2}$


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

%% CORRECT
$\left\{\bmatrix{-1\cr\hfill1\cr\hfill1\cr\hfill0},
  \bmatrix{-4\cr\hfill2\cr\hfill0\cr\hfill1}\right\}$
%% WRONG
$\left\{\bmatrix{\hfill1\cr\hfill2\cr-1\cr\hfill0},
  \bmatrix{\hfill0\cr\hfill1\cr-1\cr-2}\right\}$
%% WRONG
$\left\{\bmatrix{1\cr0\cr0},\bmatrix{2\cr1\cr0},\bmatrix{-1\cr-1\cr0},
  \bmatrix{\hfill0\cr-2\cr\hfill0}\right\}$
%% WRONG
$\left\{\bmatrix{\hfill1\cr-3\cr\hfill2},
  \bmatrix{\hfill2\cr-4\cr\hfill5}\right\}$
%% WRONG
$\left\{\bmatrix{1\cr0\cr0\cr0},\bmatrix{0\cr1\cr0\cr0},
  \bmatrix{0\cr0\cr0\cr1} \right\}$


%10
%% PROBLEM
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 $W$.

%% CORRECT
2
%% WRONG
1
%% WRONG
3
%% WRONG
4
%% WRONG
5


%11
%% PROBLEM
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~$B$.

%% CORRECT
$\bmatrix{\hfill2\cr-1\cr\hfill3\cr\hfill4}$
%% WRONG
$\bmatrix{\hfill4\cr\hfill3\cr-1\cr\hfill2}$
%% WRONG
$\bmatrix{\hfill2\cr-1\cr\hfill5\cr\hfill3}$
%% WRONG
$\bmatrix{\hfill2\cr\hfill5\cr-1\cr\hfill3}$
%% WRONG
$\bmatrix{1\cr0\cr1\cr0}$


%12
%% PROBLEM
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.

%% CORRECT
$\rank A = 2$
%% WRONG
$\det A \ne 0$
%% WRONG
$A$ is invertible.
%% WRONG
$\rank A = 1$
%% WRONG
$A$ must be symmetric.


%13
%% PROBLEM
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 $W(y_1,y_2)(t)$.

%% CORRECT
$t$
%% WRONG
$e^{t^2}$
%% WRONG
$1/t$
%% WRONG
$e^{1-t}$
%% WRONG
\emph{cannot be determined}


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

%% CORRECT
$y = (1-t)e^{3t}$
%% WRONG
$y = e^{3t}$
%% WRONG
$y = e^{3t}-t$
%% WRONG
$y = e^{-3t} + 5t e^{-3t}$
%% WRONG
$y = e^t\cos(3t) + e^t \sin(3t)$


%15
%% PROBLEM
Find the general solution of $y'' - 4y' + 13y = 0$.

%% CORRECT
$c_1 e^{2t}\cos(3t) + c_2 e^{2t}\sin(3t)$
%% WRONG
$c_1 e^{3t}\cos(2t) + c_2 e^{3t}\sin(2t)$
%% WRONG
$c_1 \cos(2t) + c_2 \sin(2t)$
%% WRONG
$c_1 e^{2t}+ c_2 e^{-2t}$
%% WRONG
$c_1 e^{5t}+ c_2 e^{-t}$


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

%% CORRECT
$\big( 2 e^{6-3x^{2}/2} - 1 \big)^{1/3}$
%% WRONG
$e^{-2 + x^{2}/2}$
%% WRONG
$\big( 2 e^{6-3x^{2}/2} \big)^{1/3} - 1 $
%% WRONG
$\big( e^{-3x^{2}/2} - 1 \big)^{1/3}$
%% WRONG
$e^{2 - x^{2}/2}$

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

%% CORRECT
$e^{2y}$
%% WRONG
$2y$
%% WRONG
$e^{-2x}$
%% WRONG
$\frac{1}{x}$
%% WRONG
$x e^{2y}$

%18
%% PROBLEM
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.

%% CORRECT
$200 - 200 e^{-0.03t}$
%% WRONG
$e^{6t} -1$
%% WRONG
$6t$
%% WRONG
$100 e^{-6t} - 100$
%% WRONG
$200 e^{0.03t} -200$


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

%% CORRECT
$y_p = e^{-t}\big(A\sin(t) + B\cos(t) + C t^2 + D t^3\big)$
%% WRONG
$y_p = t^2e^{-t}\big(A\sin(t) + B\cos(t) + C + D t\big)$
%% WRONG
$y_p = e^{-t}\big(A\sin(t) + B\cos(t) + C t^3\big)$
%% WRONG
$y_p = t^2e^{-t}\big(A\sin(t) + B\cos(t) + C\big)$
%% WRONG
$y_p = e^{-t}\big(A\sin(t) + B\cos(t) + C + D t\big)$


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

%% CORRECT
$y_p = e^t\big(t - t^2/2\big)$
%% WRONG
$y_p = e^t(1 - t)$ 
%% WRONG
$y_p = e^{2t} + te^t$
%% WRONG
$y_p = t e^t + t^2$
%% WRONG
$y_p = e^t(1-t)^3/3$


%% END
\bye
