\magnification=\magstep1
\hsize=6.5truein
\vsize=8.5truein

\parindent=0pt

\def\u{{\bf u}}
\def\v{{\bf v}}
\def\w{{\bf w}}
\def\R{{\bf R}}

\def\bmatrix#1{\left[\matrix{#1}\right]}
\def\Span{{\rm Span}\,}

\def\problemspace{0.25in}

%% TITLE
\noindent
{\bf MATH 228: Intro to Lin Alg \& Diff Eqns}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl February 12, 2002}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}

\bigskip
\noindent
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this page. There are 10 multiple
choice questions worth 5 points each and 3 partial credits problems worth
10 points each. You start with 20 points.  On the partial credit problems
{\it you must show your work and all important steps to receive credit}.

\bigskip

\def\answerspace{0.25in}


%1
%% PROBLEM
The augmented matrix of a system of equations has been reduced to row
echelon form:
$$\bmatrix{
2&-1&5&-3&1\cr
0&1&-3&1&1\cr
0&0&0&0&0\cr
0&0&0&0&0}$$
Find all solutions of the system.

%% CORRECT
$\{(-s+t+1,3s-t+1,s,t)\,|\, s,t\ {\rm real\ numbers}\}$
%% WRONG
$\{(-s+t+1,3s-t+1,0,0)\,|\, s,t\ {\rm real\ numbers}\}$
%% WRONG
$x_1=1$, $x_2=3$, $x_3=1$, $x_4=1$
%% WRONG
$\{(-r+s-t,3r-s-t+1,r,s,t)\,|\, r,s,t\ {\rm real\ numbers}\}$
%% WRONG
There are no solutions.

%2
%% PROBLEM
Let $A=\bmatrix{0&0&0&0&0\cr
-3&0&3&-8&-10\cr
1&0&-1&3&4\cr
2&0&-2&5&6}$.
Determine which of the following matrices gives $A$ reduced to row
echelon form.

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


%3
%% PROBLEM
Let $A = \bmatrix{1&3&-2&6\cr2&0&4&7}$. Determine which of the following
statements is true.

%% CORRECT
If $B = \bmatrix{1\cr2\cr3\cr4}$, then $AB$ is defined and is a
$2\times1$ matrix.

%% WRONG
If $B = \bmatrix{1&2&3&0\cr-1&1&2&4}$, then $AB$ is defined and is a
$2\times4$ matrix.

%% WRONG
If $B = \bmatrix{1&2\cr0&4}$, then $AB$ is defined and is a $2\times4$
matrix.

%% WRONG
If $B = \bmatrix{1\cr2\cr3\cr4}$, then $AB$ is defined and is a
$4\times1$ matrix.

%% WRONG
If $B = \bmatrix{1&-1\cr2&1\cr3&2\cr4&0\cr}$, then $AB$ is defined and is a
$4\times2$ matrix.


%4
%% PROBLEM
Let $A= \bmatrix{1&1&1\cr-2&-1&0\cr1&2&2}$. Find $A^{-1}$.

%% CORRECT
$A^{-1} = \bmatrix{2&0&-1\cr-4&-1&2\cr3&1&-1}$
%% WRONG
$A^{-1} = \bmatrix{-1&0&-1\cr3&1&0\cr-1&-1&1}$
%% WRONG
$A^{-1} = \bmatrix{3&1&-1\cr-4&-1&2\cr2&0&-1}$
%% WRONG
$A^{-1} = \bmatrix{1&0&0\cr2&1&0\cr3&1&-1}$
%% WRONG
$A$ is not invertible.

%5
%% PROBLEM
 Let $A$ be an $n \times n$ matrix. Which of the following
statements are NOT correct?

\halign{\hfil#\hfil&#\hfil\cr
  (I) & $\det A = \det A^{T}$\cr
 (II) & $\det(AB) = (\det A) (\det B)$\cr
(III) & $\det(A+B) = \det A + \det B$\cr
 (IV) & $\det (2A) = 2^{n} \det A $\cr
  (V) & If $A$ is not invertible, then $\det A =0$.\cr
}

%% CORRECT
(III) only
%% WRONG
(V) only
%% WRONG
(IV) only
%% WRONG
(III) and (IV)
%% WRONG
(I) and (V)


%6
%% PROBLEM
Compute $\det \bmatrix{
1&2&1&-1\cr
-1&-2&-1&2\cr
2&4&1&-2 \cr
3&2&3&-3}$.

%% CORRECT
$-4$
%% WRONG
$0$
%% WRONG
$-1$
%% WRONG
$6$
%% WRONG
$4$


%7
%% PROBLEM
Find the $(2,3)$-cofactor of $A = \bmatrix{1&-2&0\cr2&4&-3\cr5&3&9}$.

%% CORRECT
$-13$
%% WRONG
$3$
%% WRONG
$-7$
%% WRONG
$18$
%% WRONG
$-6$

%8
%% PROBLEM
Let $A = \bmatrix{2&-1&0\cr-1&2&-1\cr0&-1&2}$, $X = \bmatrix{x_1\cr
x_2\cr x_3\cr}$, and $B = \bmatrix{1\cr-1\cr2}$.  If $X$ is the solution
of $AX = B$, determine which of the following expressions gives $x_3$,
according to Cramer's Rule.

%% CORRECT
${1\over4}\det\bmatrix{2&-1&1\cr-1&2&-1\cr0&-1&2}$
%% WRONG
${1\over4}\det\bmatrix{2&1&0\cr-1&-1&-1\cr0&2&2}$
%% WRONG
${1\over4}\det\bmatrix{1&-1&0\cr-1&2&-1\cr2&-1&2}$
%% WRONG
${1\over4}\det\bmatrix{2&-1&0\cr-1&2&-1\cr0&-1&2}$
%% WRONG
${1\over4}\det\bmatrix{2&-1&0\cr-1&1&-1\cr0&-1&2}$


%9
%% PROBLEM
Let $\u = (2,-2,0,0)$, and $\v = (3,0,0,-3)$. Find the angle between $\u$ and
$\v$.

%% CORRECT
$\pi/3$
%% WRONG
$\pi/2$
%% WRONG
$\pi/4$
%% WRONG
$\pi/6$
%% WRONG
$3\pi/4$

%10
%% PROBLEM
Let $\u = (1,1,0)$, $\v = (1,0,-1)$, and $\w = (-1,-1,0)$. Determine
which of the following describes $\Span\{\u,\v,\w\}$.

%% WRONG
point
%% WRONG
line
%% CORRECT
plane
%% WRONG
$R^3$
%% WRONG
none of the above
%% STET

%% END

\vfill\eject

\noindent\llap{11.\ }%
Find all solutions of the system
$$\vbox{\halign{\hfil$#$&$\ #\ $&\hfil$#$&$\ #\ $&\hfil$#$&$\ #\ $&\hfil$#$\cr
  x_1 &+& 2x_2 &-&  x_3 &=&  3\cr
 2x_1 &+& 4x_2 &-&  x_3 &=&  4\cr
  x_1 &+& 3x_2 & &      &=&  5\cr
-3x_1 &-& 5x_2 &+& 3x_3 &=& -5\cr
}}$$

\vfill\eject

\noindent\llap{12.\ }%
Find the $LU$ decomposition of the matrix $A = \bmatrix{1&1&1\cr
-1&2&0\cr 2&11&5}$.


\vfill\eject

\noindent\llap{13.\ }%
Let $V$ be the vector space of polynomials of degree $\le 2$ with inner
product defined for $f,g\in V$ by
$$ f\cdot g = f(0)g(0) + f'(0)g'(0) + {1\over4}f''(0)g''(0)$$
Compute the projection of $f(x) = 1 + x + x^2$ on $g(x) = x^2 - 2$.

\bye
