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%% TITLE

\noindent
\textbf{MATH 228C: Intro to Linear Alg and Diff Eqns}\hfill
Name:\makebox[2.5in][l]{\rule{2in}{.5pt}}

\medskip\noindent
\textbf{Final Exam}\quad{\sl May 6, 2004}

\vspace{.5in}
\noindent

There are 20 multiple choice questions worth 7 points each for a total of
140 points. You start with 10 points.  Record your answers by placing an
$\times$ through one letter for each problem on this page.

\medskip\noindent
\fbox{Calculators are \emph{not} allowed.}

\def\answerspace{0.25in}

%% PROBLEM 1

Find the general solution of the linear system
\[ \begin{array}{rcrcrcrcl}
x_1 & - & x_2 & + & 2x_3 & + &  x_4 & = & 1\\
    &   & x_2 & + &  x_3 & - &  x_4 & = & 0\\
x_1 &   &     & + &  x_3 & + & 2x_4 & = & 3
\end{array} \]

%% CORRECT

$x_1 = 4 - 3t$, $x_2 = 1$, $x_3 = -1+t$, $x_4 = t$

%% WRONG

$x_1 = 4 $, $x_2 = 1$, $x_3 = -1$, $x_4 = 0$

%% WRONG

$x_1 = 4 - t$, $x_2 = 1$, $x_3 = -1-t$, $x_4 = t$

%% WRONG

$x_1 = 1 + s - 3t$, $x_2 = s$, $x_3 = -1+t$, $x_4 = t$

%% WRONG

$x_1 = s - t$, $x_2 = -1 + s $, $x_3 = -1-t$, $x_4 = t$


%% PROBLEM 2

Let $A = \left[\begin{array}{cc} 3 & 2\\ 1 & 1\end{array}\right]$
and $B = \left[\begin{array}{ccc} 1 & 2 & 3\\ -1 & 0  & 2\end{array}\right]$.
Compute the matrix product $A^{-1}B$.

%% CORRECT

$\left[\begin{array}{ccc} 3 & 2 & -1\\ -4 & -2  & 3\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc} 1 & 6 & 13\\ 0 & -2  & 5\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc} -1 & 2 & 4\\ -3 & -1  & 2\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc} 2 & 3 & -1\\ -4 & 2  & -1\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc} 1 & 13 & 6\\ 0 & 5  & -2\end{array}\right]$

%% PROBLEM 3

Determine which of the following is an inner product on the vector space of
$2\times 2$ matrices.

%% CORRECT

$\langle A, B\rangle = \tr(A^TB)$

%% WRONG

$\langle A, B\rangle = \det(AB)$

%% WRONG

$\langle A, B\rangle = \det(A+B)$

%% WRONG

$\langle A, B\rangle = \tr(A+B)$

%% WRONG

$\langle A, B\rangle = A^TB$



%% PROBLEM 4

Compute the determinant of the matrix
$\left[\begin{array}{cccc}
1 & 2 & 3 & 4\\
2 & 1 & 0 & 0\\
3 & 0 & 1 & 0\\
4 & 0 & 0 & 1
\end{array}\right]$.

%% CORRECT

$-28$

%% WRONG

$24$

%% WRONG

$-18$

%% WRONG

0

%% WRONG

$12$


%% PROBLEM 5

Find a basis for $\span\{(1,-1,1,0),(-1,1,0,1),(1,-1,2,1)\}$.

%% CORRECT

$\{(1,-1,1,0), (0,0,1,1)\}$

%% WRONG

$\{(1,-1,1,0), (-1,0,1,1)\}$

%% WRONG

$\{(1,-1,1,0), (0,0,2,1), (0,0,0,1)\}$

%% WRONG

$\{(1,-1,1,0)\}$

%% WRONG

$\{(1,0,0,0), (0,1,0,0), (0,0,1,0)\}$


%%PROBLEM 6

Let $V$ be the vector subspace of continuous functions with basis
$B = \{\cos(x), \sin(x), x \cos(x), x\sin(x)\}$. Find the matrix of the
linear transformation $T:V\to V$ defined by $T(f(x)) = f'(x)$  with respect
to $B$.

%% CORRECT

$\left[\begin{array}{rrrr}
0 & 1 & 1 & 0\\
-1 & 0 & 0 & 1\\
0 & 0 & 0 & 1\\
0 & 0 & -1 & 0
\end{array}\right]$

%% WRONG

$\left[\begin{array}{rrrr}
0 & 1 & 1 & 0\\
-1 & 0 & 0 & 1\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & 1
\end{array}\right]$


%% WRONG

$\left[\begin{array}{rrrr}
0 & 0 & 1 & 0\\
0 & 1 & 0 & 1\\
-1 & 0 & -1 & 0\\
0 & 0 & 0 & 1
\end{array}\right]$


%% WRONG

$\left[\begin{array}{rrrr}
1 & 0 & 0 & 1\\
0 & -1 & 1 & 0\\
0 & 0 & 0 & 1\\
0 & 0 & -1 & 0
\end{array}\right]$


%% WRONG

$\left[\begin{array}{rrrr}
0 & 1 & 1 & 0\\
-1 & 0 & 0 & 1\\
 0 & 1 & 0 & 0\\
-1 & 0 & 0 & 0
\end{array}\right]$



%% PROBLEM 7

The Gram-Schmidt process is applied to the following basis of $\R^3$: $u_1
= (2,2,1)$, $u_2 = (-2,1,2)$, $u_3 = (0,0,2)$, resulting in the orthonormal
vectors $q_1 = (2/3,2/3,1/3)$, $q_2 = (-2/3,1/3,2/3)$, and $q_3$. Find
$q_3$.

%% CORRECT

$(1/3,-2/3,2/3)$

%% WRONG

$(1/3,2/3,-2/3)$

%% WRONG

$(2/3,-1/3,-2/3)$

%% WRONG

$(0,0,1)$

%% WRONG

$(2/3,2/3,1/3)$


%% PROBLEM 8

Find the line $y = ax + b$ that is the best least-squares fit through the
points $(1,1)$, $(1,2)$ and $(2,3)$.

%% CORRECT

$y = \frac{3}{2}x$

%% WRONG

$y = \frac{5}{4}x + \frac{1}{8}$

%% WRONG

$y = \frac{5}{2}x - \frac{1}{4}$

%% WRONG

$y = \frac{4}{3}x + \frac{1}{8}$

%% WRONG

$y = x + \frac{1}{2}$


%% PROBLEM 9

A matrix $A$ has eigenvectors
$\left[\begin{array}{c}1\\0\\0\end{array}\right]$,
$\left[\begin{array}{c}1\\1\\0\end{array}\right]$, and
$\left[\begin{array}{c}1\\0\\1\end{array}\right]$ with corresponding
eigenvalues 1, 2, and 3. Use this information to reconstruct the matrix~$A$.

%% CORRECT

$\left[\begin{array}{ccc}1 & 1 & 2\\ 0 & 2 & 0\\ 0 & 0 & 3\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc}1 & 2 & 3\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc}1 & 1 & 0\\ 0 & 2 & 1\\ 0 & 0 & 3\end{array}\right]$

%% WRONG

$\left[\begin{array}{ccc}1 & 1 & 1\\ 0 & 2 & 1\\ 0 & 0 & 3\end{array}\right]$


%% PROBLEM 10

Suppose $B = \{v_1, v_2, v_3\}$ and $C = \{u_1,u_2,u_3\}$ are bases of a
vector space $V$ related by $v_1 = u_1 - u_3$, $v_2 = u_2 - u_3$, $v_3 =
u_3 - u_2$.  If $[x]_B = \left[\begin{array}{c}2\\1\\-1\end{array}\right]$,
find $[x]_C$.

%% CORRECT

$\left[\begin{array}{c}2\\2\\-4\end{array}\right]$

%% WRONG

$\left[\begin{array}{c}3\\2\\-2\end{array}\right]$

%% WRONG

$\left[\begin{array}{c}1\\2\\-1\end{array}\right]$

%% WRONG

$\left[\begin{array}{c}2\\1\\-3\end{array}\right]$

%% WRONG

$\left[\begin{array}{c}1\\1\\-2\end{array}\right]$


%% PROBLEM 11

Suppose $y(t)$ satisfies the initial value problem $y' - \tan(t)y =
\sec(t)$, $y(0) = 0$. Find $y(\pi/4)$.

%% CORRECT

$\pi\sqrt{2}/4$

%% WRONG

$\pi/4$

%% WRONG

$\pi\sqrt{2}/8$

%% WRONG

$\pi/8$

%% WRONG

$0$


%% PROBLEM 12

Find an integrating factor for $(x^3 + xy^3 + y^3)\,dx + 3xy^2\,dy = 0$.

%% CORRECT

$e^x$

%% WRONG

$3xy^2$

%% WRONG

$e^y$

%% WRONG

$3x^2 y$

%% WRONG

$x$


%% PROBLEM 13

Suppose the air resistance on a high-speed 1~kg projectile moving in a
straight line is proportional to the square of its speed. The projectile
has an initial speed of 500 meters per second (mps) and the air resistance
at that speed is 25~newtons. Find the speed of the projectile after 5
seconds. (Ignore other forces.)

%% CORRECT

400 mps

%% WRONG

250 mps

%% WRONG

300 mps

%% WRONG

350 mps

%% WRONG

200 mps


%% PROBLEM 14

Solve the initial value problem $y'' - 2y' + y = 0$, $y(1) = 0$, $y'(1) = 1$.

%% CORRECT

$(t-1)e^{t-1}$

%% WRONG

$t e^t$

%% WRONG

$e^t - t e^t$

%% WRONG

$(t-1)e^t/2$

%% WRONG

$(te^{t-1}-1)/2$


%% PROBLEM 15

Find the general solution of $y'' -2y' - 3y = 0$.

%% CORRECT

$c_1e^{-t} + c_2e^{3t}$

%% WRONG

$c_1e^{t} + c_2e^{-3t}$

%% WRONG

$c_1e^{-2t} + c_2e^{-3t}$

%% WRONG

$c_1e^{2t} + c_2e^{3t}$

%% WRONG

$e^{-t}(c_1 + c_2t)$


%% PROBLEM 16

Compute the Wronskian of $y_1 = t\sin(t)$ and $y_2 = t\cos(t)$.

%% CORRECT

$-t^2$

%% WRONG

$-2t\sin(t)\cos(t)$

%% WRONG

$-t^2\sin(t)\cos(t)$

%% WRONG

$-t^2(\sin^2(t)-\cos^2(t))$

%% WRONG

$-1$


%% PROBLEM 17

Let $L$ be the linear operator $L[y] = y'' + p(t)y' + q(t)y$.
Given that $\{t+1, e^t\}$ is a basis for the kernel of $L$,
find a function $\phi(t)$ that satisfies $L[\phi(t)] = te^t$.

%% CORRECT

$t^2e^t/2$

%% WRONG

$t^2/2+t-e^t$

%% WRONG

$(t^2/2+t-1)e^t$

%% WRONG

$(t-1)e^t$

%% WRONG

$(t^2-1)e^t$


%% PROBLEM 18

Find a particular solution to $y'' + y' + y = t^3$.

%% CORRECT

$t^3 - 3t^2 + 6$

%% WRONG

$e^{-t/2}(\cos(\sqrt{3}t) + \sin(\sqrt{3}t)) + t^3/6$

%% WRONG

$e^{-t/2}(\cos(\sqrt{3}t) - \sin(\sqrt{3}t))/\sqrt{3} + t^3-3t^2$

%% WRONG

$e^{-t/2}(2\cos(\sqrt{3}t) + \sin(\sqrt{3}t)/\sqrt{3})$

%% WRONG

$(3t^2-t^3)/6$


%% PROBLEM 19

Suppose a spring-mass system is governed by the differential equation
$2u'' +  4u' + 3u = 0$ with initial conditions $u(0) = 1$, $u'(0) = 0$.
Determine which of the following adjectives best describes the system.

%% CORRECT

oscillating

%% WRONG

overdamped

%% WRONG

undamped

%% WRONG

critically damped

%% WRONG

resonating


%% PROBLEM 20

Determine which of the following is a \emph{stable} equilibrium of the
autonomous system $\frac{dy}{dt} = 2y - 3y^2 + y^3$.

%% CORRECT

$1$

%% WRONG

$2$

%% WRONG

$3$

%% WRONG

$4$

%% WRONG

$0$

%% END

\end{document}
