\magnification=\magstep1
\hsize=5.5in
\vsize=7.5in

\parindent=0pt

\def\q{{\bf q}}
\def\u{{\bf u}}
\def\v{{\bf v}}
\def\x{{\bf x}}
\def\R{{\bf R}}

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

\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 Exam II}\quad{\sl March 19, 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
Let $\v_1 = (1, 1, 0)$, $\v_2 = (1, 0, 1)$, $\v_3 = (-1, c, -3)$. Find
a value of $c$ such that $\v_1$, $\v_2$ and $\v_3$ are linearly dependent.

%% CORRECT
2
%% WRONG
0
%% WRONG
$-2$
%% WRONG
1
%% WRONG
$-3$


%2
%% PROBLEM
Determine which of the following sets of polynomials forms a basis
of $P_2$, the vector space of all polynomials of degree less than or
equal to~2.

%% CORRECT
$\{ 1+x-x^2, x+x^2, 1+x+x^2\}$
%% WRONG
$\{1+71x+23x^2, 2x+59x^2\}$
%% WRONG
$\{1+2x+x^2, 2-x+x^2, 3+x+2x^2\}$
%% WRONG
$\{1+x, 1+2x+x^2, 3+4x+5x^2, 8+7x+2x^2\}$
%% WRONG
$\{3-x+x^2, 5x+x^2, 1+2x+3x^2, 1+4x+5x^2\}$


%3
%% PROBLEM
What is the dimension of the space spanned by
$\bmatrix{ 1& -1\cr 1&  0}$,
$\bmatrix{-1& 0\cr -2&  1}$,
$\bmatrix{ 1& -3\cr -1&  2}$.

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


%4
%% PROBLEM
Determine the rank of the matrix $A = \bmatrix{
 2 &  6 & -1 &  1 \cr
-1 & -3 &  1 &  0 \cr
-3 & -9 &  2 & -1 \cr
 4 & 12 & -2 &  2 }$.

%% CORRECT
2
%% WRONG
0
%% WRONG
1
%% WRONG
3
%% WRONG
4

%5
%% PROBLEM
Determine which of the following sets are vector spaces with usual addition
and scalar multiplication.

(I) Set of all continuous functions $f$ on $[0, 1]$ with
$f(0)=0$ and $f(1)=1$.

(II) Set of all $2 \times 2$ matrices of the form
$\bmatrix{0 & a \cr b & c }$
where $a$, $b$, $c$ are arbitrary real numbers.

(III) Set of all $2 \times 2$ matrices of the form
$\bmatrix{ a & 1 \cr 0 & b}$
where $a, b$ are arbitrary real numbers.

(IV) Set of all polynomials $p(x) = a_0 + a_1 x + a_2 x^2$
with $a_0 + a_2 =0$.

(V) Set of all $2 \times 2$ matrices $A$ with $\det A = 0$.

%% CORRECT
(II) and (IV)
%% WRONG
(I) and (II)
%% WRONG
(IV) and (V)
%% WRONG
(I) and (III)
%% WRONG
(II), (IV) and (V)


%6
%% PROBLEM
Find the coordinates, $[\x]_B$ of the vector $\x=(1,0,-1)$ with respect to
the basis $B = \{(1,-1,0), (-1,1,-1), (0,-1,1)\}$.
%% CORRECT
$\bmatrix{1\cr0\cr-1}$
%% WRONG
$\bmatrix{1\cr1\cr-1}$
%% WRONG
$\bmatrix{2\cr-1\cr0}$
%% WRONG
$\bmatrix{1\cr0\cr-2}$
%% WRONG
$\bmatrix{1\cr-1\cr-1}$


%7
%% PROBLEM
Suppose $B = \{\u_1,\u_2,\u_3\}$ and $C = \{\v_1,\v_2,\v_3\}$ are bases of a
vector space $V$ related by
$$\eqalign{
\u_1 &= \v_1 - \v_3\cr
\u_2 &= \v_2 - \v_3\cr
\u_3 &= \v_1 - \v_2}$$
If $[x]_B =\bmatrix{1\cr2\cr3}$, find $[x]_C$.

%% CORRECT
$\bmatrix{4\cr-1\cr-3}$
%% WRONG
$\bmatrix{-2\cr-1\cr-1}$
%% WRONG
$\bmatrix{-1\cr0\cr2}$
%% WRONG
$\bmatrix{-1\cr4\cr2}$
%% WRONG
$\bmatrix{3\cr-1\cr-2}$


%8
%% PROBLEM
The Gram-Schmidt process is applied to the following basis of $\R^3$:
$$\v_1 = \bmatrix{2\cr2\cr1},\quad
\v_2 = \bmatrix{-2\cr1\cr2},\quad
\v_3 = \bmatrix{18\cr0\cr0}$$
resulting in the orthonormal vectors
$$\q_1 = \bmatrix{2/3\cr2/3\cr1/3},\quad
\q_2 = \bmatrix{-2/3\cr1/3\cr2/3},\quad
\q_3$$
Find $\q_3$.

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


%9
%% PROBLEM
Let $V = C[0,1]$ be the vector space of continuous functions on $[0,1]$
with inner product defined by $f\cdot g = \int_0^1 f(x)g(x)\,dx$. Let
$f(x) = -6x + 2$. Find the number $a$ such that $g(x) = x+a$ is orthogonal
to~$f$.

%% CORRECT
$-1$
%% WRONG
0
%% WRONG
$-3$
%% WRONG
2
%% WRONG
$1/3$

%10
%% PROBLEM
Find $R$ in the $QR$-decomposition of the matrix $A =
\bmatrix{4&25\cr0&0\cr3&-25}$.
%% CORRECT
$\bmatrix{5&5\cr0&35}$
%% WRONG
$\bmatrix{5&0\cr15&35}$
%% WRONG
$\bmatrix{25&0\cr0&25}$
%% WRONG
$\bmatrix{4&3\cr0&25}$
%% WRONG
$\bmatrix{12&-4\cr0&25}$


%% END

\vfill\eject

\noindent\llap{11.\ }%
Find a basis for the null space, $NS(A)$, of the matrix
$A=  \left[\matrix{
 1 & -1 &  1 & 1 \cr
 2 & -1 &  2 & 1 \cr
-1 &  0 & -1 & 0 \cr
 3 & -2 &  3 & 2}\right]$.

\vfill\eject


\noindent\llap{12.\ }%
Define $T:C[0,1]\rightarrow C[0,1]$ as follows: If $f$ is a continuous
function on $[0,1]$, then $T(f)$ is the continuous function of $x$
defined by $\int_0^x t f(t)\,dt$ for $0 \le x \le 1$. For example, if $f$
is the function $f(x)=x^2$, then $T(f)$ is the function $\int_0^x t\cdot
t^2\,dt = x^4/4$. Determine whether $T$ is a linear transformation. Be
sure to justify the statements leading to your conclusion.

\vfill\eject


\noindent\llap{13.\ }%
Let $A = \bmatrix{2&3&6\cr6&2&-3\cr3&-6&2}$.

(a) Find a scalar $c$ so that $cA$ is an orthogonal matrix.

\vfill

(b) Use your answer in part (a) to find $A^{-1}$.

\vfill

\bye
