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

\def\nullity{{\rm nullity}}
\def\rank{{\rm rank}}
\def\trace{{\rm trace}}
\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\emph#1{{\it #1}}
\def\frac#1#2{{{#1}\over{#2}}}
\def\b{{\bf b}}
\def\R{{\bf R}}
\def\v{{\bf v}}
\def\x{{\bf x}}
\def\0{{\bf 0}}

\def\problemspace{0.75in}

%% TITLE
\noindent
\leavevmode\hfill Name:\hbox to 2.5truein{\hrulefill}

\medskip

\leavevmode\hfill Instructor:\hbox to 2.5truein{\hrulefill}

\bigskip

\centerline{\bf MATH 228: Introduction to Linear Algebra and Differential
Equations}

\medskip

\centerline{\bf Final Exam}

\medskip

\centerline{\sl May 7, 2003}

\bigskip

\noindent
Record your answers by placing an $\times$ through one letter for each
problem on this page. There are 21 questions worth 6 points each. You
start with 24 points.

\medskip

\noindent
You may \emph{not} use a calculator.

\medskip

\def\answerspace{0.25in}

%1
%% PROBLEM
Determine the number of parameters in the general solution of the linear
system
$$\vbox{
\halign{&\hfil$~#~$\cr
x &+&  y &+& z &+& w &=& 0\cr
x &+& 2y & &   & &   &=& 1\cr
  & &  y &-& z &-& w &=& 1\cr
}}$$

%% CORRECT
2
%% WRONG
0
%% WRONG
1
%% WRONG
3
%% WRONG
none, it is inconsistent


%2
%% PROBLEM
Compute the determinant of
$\bmatrix{
0&1&2&3\cr
1&2&3&0\cr
2&3&0&1\cr
3&0&1&2}$.

%% CORRECT
96
%% WRONG
$-16$
%% WRONG
0
%% WRONG
48
%% WRONG
$-72$


%3
%% PROBLEM
Let $T$ be a linear transformation $T:\R^3\to\R^3$ that satisfies
$T(1,0,1) = (0,1,0)$, $T(0,1,1)=(1,0,0)$, and $T(1,1,0) = (0,0,1)$.
Find $T^{-1}(1,2,3)$.
%% CORRECT
$(5,4,3)$
%% WRONG
$(3,4,5)$
%% WRONG
$(4,5,3)$
%% WRONG
$(3,5,4)$
%% WRONG
$(4,3,5)$



%4
%% PROBLEM
Find the first coordinate of the vector $(5,6,7)$ with respect to the basis
$\{(1,0,1), (1,0,2), (0,-1,0)\}$.
%% CORRECT
3
%% WRONG
2
%% WRONG
$-6$
%% WRONG
$-1$
%% WRONG
5

%5
%% PROBLEM
Determine which of the following sets is \emph{not}
subspace of the given vector space $V$.
%% CORRECT
All $A$ in $M_{n,n}$ such that $A^{-1} = -A$.
%% WRONG
All polynomials $a_0 + a_1x + a_2x^2 + a_3x^3$ in $P_3$ such that
$a_3=a_0+a_1+a_2$.
%% WRONG
All $A$ in $M_{n,n}$ such that $A^T = -A$.
%% WRONG
All functions $f(x)$ in $C[0,1]$ such that $f(1) = 0$.
%% WRONG
All functions $f(x)$ in $C[0,1]$ such that $\int_0^1\!\!\!f(x)dx = 0$.


%6
%% PROBLEM
Determine which of the following statements is true about the
functions $\sin(x)$, $\cos(-x)$, $\sin(x-\pi/2)$.

%% CORRECT
They are linearly dependent since their span is two-dimensional.
%% WRONG
They are linearly independent because $\sin(x)$ is not a multiple of
$\cos(-x)$.
%% WRONG
They are linearly independent by the Wronskian test.
%% WRONG
They are linearly dependent since $\cos(-x) = \cos(x)$.
%% WRONG
They are linearly dependent since their kernel is non-zero.


%7
%% PROBLEM
Find a basis for the column space of the matrix
$\bmatrix{
 1&-1& 2& 1\cr
-1& 1&-1& 0\cr
 2&-2& 1&-1}$.

%% CORRECT
$\bmatrix{1\cr-1\cr2}$, $\bmatrix{2\cr-1\cr1}$
%% WRONG
$\bmatrix{1\cr-1\cr2}$, $\bmatrix{2\cr-1\cr1}$, $\bmatrix{1\cr0\cr-1}$
%% WRONG
$\bmatrix{1\cr0\cr0}$, $\bmatrix{2\cr1\cr0}$
%% WRONG
$\bmatrix{1\cr-1\cr2}$, $\bmatrix{-1\cr1\cr1}$
%% WRONG
$\bmatrix{1\cr0\cr0}$, $\bmatrix{2\cr1\cr0}$, $\bmatrix{1\cr1\cr-3}$


%8
%% PROBLEM
Determine the rank and nullity of the matrix
$A=  \bmatrix{
1&2&3&4\cr
0&2&3&4\cr
0&0&3&4\cr
1&2&3&4}$.

%% CORRECT
$\rank(A)=3$, $\nullity(A)=1$
%% WRONG
$\rank(A)=1$, $\nullity(A)=3$
%% WRONG
$\rank(A)=2$, $\nullity(A)=2$
%% WRONG
$\rank(A)=3$, $\nullity(A)=3$
%% WRONG
$\rank(A)=1$, $\nullity(A)=1$


%9
%% PROBLEM
A symmetric $3\times 3$ matrix $A$ has characteristic equation
$(\lambda-1)(\lambda-2)^2 =0$. Determine which of the following statements
is true.

%% CORRECT
$A$ is orthogonally diagonalizable.
%There exists an orthogonal martix $P$ such that
%$P^{-1}AP= \bmatrix{
%2&0&0\cr
%0&1&0\cr
%0&0&2}$

%% WRONG
The $\lambda=1$ eigenspace has dimension $1$ but the $\lambda=2$
eigenspace could have dimension $1$ or $2$.
%% WRONG
$A$ is diagonalizable but may not necessarily be orthogonally
diagonalizable.
%% WRONG
$A$ is not necessarily diagonalizable.
%% WRONG
$A$ must be a diagonal matrix.


%10
%% PROBLEM
A $3\times 3$ matrix $A$ has eigenvalues $\lambda=1$ and $\lambda=2$.
A basis for the eigenspace corresponding to $\lambda=1$ is
$\bmatrix{1\cr0\cr0}$.
A basis for the eigenspace corresponding to $\lambda=2$ is
$\bmatrix{1\cr1\cr 1}$, $\bmatrix{1\cr-1\cr0}$.
Determine which of the following is the matrix~$A$.

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


%11
%% PROBLEM
Find the standard matrix for the orthogonal projection of $\R^3$ onto the
subspace spanned by $\bmatrix{1\cr1\cr1}$ and $\bmatrix{1\cr0\cr0}$.

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


%12
%% PROBLEM
Determine which of the following is an orthogonal matrix.

%% CORRECT
$\bmatrix{
0&0&0&1\cr
0&0&1&0\cr
0&1&0&0\cr
1&0&0&0}$
%% WRONG
$\bmatrix{1&-1\cr0&1}$ 
%% WRONG
$\bmatrix{
1&-1/\sqrt{2}& 1/\sqrt{3}\cr
0& 1/\sqrt{2}& 1/\sqrt{3}\cr
0& 0& 1/\sqrt{3}}$
%% WRONG
$\bmatrix{
1& 0& 1& 0\cr
0& 1& 0& 1\cr
1& 0&-1& 0\cr
0& 1& 0&-1}$
%% WRONG
$\bmatrix{
 0&2/\sqrt{6}&-1/\sqrt{3}\cr
 1&1/\sqrt{6}& 1/\sqrt{3}\cr
-1&1/\sqrt{6}& 1/\sqrt{3}}$

%13
%% PROBLEM
A linear transformation $T:P_2 \to P_2$ is given by $T(p(x))=p(x+1)-p(x)$.
Find the standard matrix $[T]_B$ for $T$ with respect to the basis
B=$\{1,x,x^2-5\}$ of $P_2$.

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

%14
%% PROBLEM
A linear transformation $T: M_{2,2} \to  R^2$ has standard matrix
$[T]_{B',B} = \bmatrix{1&3&1&-1\cr 0&0&2&4}$ relative to the bases $B
=\left\{
\bmatrix{1&0\cr0&0},
\bmatrix{0&1\cr0&0},
\bmatrix{0&0\cr1&0},
\bmatrix{0&0\cr0&1}
\right\}$
of $M_{2,2}$ and $B'=\{ (1,0), (0,1)\}$ of $\R^2$.

Determine which of the following matrices $A$ satisfy $T(A) =
\bmatrix{1\cr2}$.

%% CORRECT
$\bmatrix{0&1\cr-1&1}$
%% WRONG
$\bmatrix{1&0\cr1&0}$
%% WRONG
$\bmatrix{0&1\cr1&-1}$
%% WRONG
$\bmatrix{0&1\cr0&0}$
%% WRONG
$\bmatrix{0&0\cr1&-1}$


%15
%% PROBLEM
Suppose a man suspended from a bungee cord stretches the cord 50~feet
longer than its natural length of 100~ft. The man jumps from a bridge while
attached to this cord and arrives at a distance of 150~ft below the bridge
with a downward velocity of 40~ft/sec. Assume the bungee cord acts like a
spring (i.e., it obeys Hooke's Law) and there are no frictional forces.
Determine how far below the bridge the man will fall before springing up again.
(The gravitational constant is $g = 32~{\rm ft}/{\rm sec}^2$.)

%% CORRECT
200 ft
%% WRONG
220 ft
%% WRONG
180 ft
%% WRONG
260 ft
%% WRONG
240 ft


%16
%% PROBLEM
The water bath surrounding a certain nuclear reactor contains
radioactive material that decays with a half-life of 4~days. New
radioactive material is added to the bath at a constant rate of
3~grams per day. Determine the amount in grams of radioactive
material in the bath after many days.

%% CORRECT
$\frac{12}{\ln{2}}$
%% WRONG
$\frac{3}{4}$
%% WRONG
$\frac{4\ln{2}}{3}$
%% WRONG
$\frac{4}{3}$
%% WRONG
$3$


%17
%% PROBLEM
The differential equation $(x^2-6xy^2+y^3)dx + (xy^2 - 3x^2y)dy = 0$

%% WRONG
is exact.
%% WRONG
is homogeneous.
%% CORRECT
has an integrating factor that is a function of $x$ alone.
%% WRONG
has an integrating factor that is a function of $y$ alone.
%% WRONG
{\sl none of the above}
%% STET


%18
%% PROBLEM
Solve the initial value problem $y'' + y' = 0$, $y(0) = 1$, $y'(0) = 0$.

%% CORRECT
$y = 1$
%% WRONG
$y = e^{-t} + t$
%% WRONG
$y = \cos(t)$
%% WRONG
$y = (e^t + e^{-t})/2$
%% WRONG
$y = e^t - t$


%19
%% PROBLEM
Let $V$ be the vector space of infinitely differentiable functions on~$\R$.
Find a basis for the kernel of the linear operator $L:V\to V$ defined by
$L[y] = 9y'' + 6y' + y$.

%% CORRECT
$\{e^{-t/3}, te^{-t/3}\}$
%% WRONG
$\{e^{t/3}, e^{-t/3}\}$
%% WRONG
$\{t, e^{-3t}\}$
%% WRONG
$\{t, e^{t/3}\}$
%% WRONG
$\{e^{3t}, t e^{3t}\}$


%20
%% PROBLEM
Find the form of a particular solution $y_p$ of $y'' - 2y' + 2y = e^t
\cos(t) + e^{-t}\sin(t)$.

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


%21
%% PROBLEM
Given that $y_1 = t$ and $y_2 = t^2$ are solutions of
$y'' + p(t)y' + q(t)y = 0$
for $t > 0$, find a particular solution $y_p$ of 
$y'' + p(t)y' + q(t)y = t^3$.

%% CORRECT
$y_p= \frac{1}{12}t^5$
%% WRONG
$y_p= -\frac{1}{4}t^4 + \frac{1}{3}t^3$
%% WRONG
$y_p= t^{-1} - \frac{1}{3}t^{-3}$
%% WRONG
$y_p= 4t^5 -3t^4+t^3$
%% WRONG
$y_p= \frac{1}{6}t^3 - t^{-1}$

%% END
\bye
