% times noreferences nopagenumbers widec tall

\par\noindent{\Large\bf Math 314/814 \kern20pt -- \kern20pt Final Exam\kern20pt 
-- \kern20pt December 12, 1994}
\vspace{0.30in}

\problemno{1} {\bf[24 points]}
For each statement below, either write {\it true}, if it holds for all
$2 \times 2$ matrices $M$, $N$, or else exhibit $2 \times 2$ matrices
$M$, $N$ (whose entries are {\it specific numbers}), for which the
statement fails.
\vspace{0.1in}
\par\noindent{\bf (a):}\ $\det(M+N) = \det(M)+\det(N)$
\par\noindent{\bf (b):}\ $\det(MN) = \det(M)\det(N)$
\par\noindent{\bf (c):}\ $MN = NM$
\par\noindent{\bf (d):}\ $M+N = N+M$

\problemno{2} {\bf[20 points]}
The system of equations 
\begin{center}
\begin{tabular}{cccccccc}
$x_1$ & $+$ & $2x_2$ &   &    &   &     &$=\ 2$\\
      &     &    & & $x_3$ & $+$ & $x_4$ &$=\ 1$\\
$2x_1$& $+$ & $4x_2$&   &    & $+$ &$2x_4$ &$=\ 6$
\end{tabular}
\end{center}
has coefficient matrix $A$ and \RHS\ $b$ such that the reduced row echelon
form of $[A\ |\ b]$ is
$$\left[ \matrixx{1&2&0&0\cr0&0&1&0\cr0&0&0&1} \left| \matrixx{2\cr0\cr1}
  \right] \right.{}.$$%
Use this information to answer the following questions:
\vspace{0.1in}
\par\noindent{\bf (a):}\ Find a basis for the solution space of $Ax = 0$.
\par\noindent{\bf (b):}\ Find the solutions of the system $Ax = b$.
\par\noindent{\bf (c):}\ Find a basis for the row space of the matrix $A$.
\par\noindent{\bf (d):}\ No matter what $b$ is, the equation $Ax = b$
has solutions.  In terms of rank, why do we know this?

\problemno{3} {\bf[10 points]}
Let $V$ be the vector space of all $2 \times 2$ matrices.  Let $W$ be the
subset of $V$ consisting of all matrices which are {\it not\/} invertible.  
Determine if $W$ is a subspace of $V$.  Justify your answer.

\problemno{4} {\bf[10 points]}
Find the coordinates of $\BRMAT{1\cr2\cr3}$ \WRT\ the basis
$$\BRMAT{3/5\cr-4/5\cr0},\ \BRMAT{4/5\cr3/5\cr0},\ \BRMAT{0\cr0\cr1}$$%
for $\R^3$.  [{\sc Hint.}\ The given basis is actually an orthonormal basis
for $\R^3$.]

\problemno{5} {\bf[15 points]}
Let $A$ be a matrix of size $3 \times 5$, and let $B$ be a matrix of size
$2 \times 3$.  Write the size in each case, or {\it undefined}:
\par\noindent{\bf (a):}\ $AB$
\par\noindent{\bf (b):}\ $BA$
\par\noindent{\bf (c):}\ $(A A^T)^2$

\problemno{6} {\bf[6 points for a, 6 points for b, 8 points for c]}

\par\noindent{\bf(a):} Let $V$ be a vector space, and let $\vec v1n \in V$.
Define: {\it $\vec v1n$ are linearly dependent}.  [Do not write
{\it not linearly independent}!]
\par\noindent{\bf(b):} Let $v_1, v_2, v_3$ be the elements of $\R^3$ given
by 
$$v_1 = \BRMAT{1\cr1\cr0},\ v_2 =\BRMAT{0\cr1\cr1},
\ \hbox{and}\ v_3 = \BRMAT{1\cr2\cr1}.$$%
Use the definition you gave in (a) to show that $v_1,v_2,v_3$ are linearly
dependent.
\par\noindent{\bf(c):}
Let $\R^3$ have the standard inner product, i.e.\ the dot product.  Find an 
orthonormal basis for the subspace of $\R^3$ spanned by $v_1$, $v_2$, and $v_3$.

\problemno{7} {\bf[12 points]}
Let $\R^2$ have the standard inner product, i.e.\ the dot product.  Consider
a subspace $W$ of $\R^2$, and a point $p \in \R^2$, as shown in the picture
below:

\widepost{190}{90}{
150 DottedAxes
-75 1.3 mul -30 1.3 mul 75 1.3 mul 30 1.3 mul SolidArrow
75 1.3 mul 30 1.3 mul -75 1.3 mul -30 1.3 mul SolidArrow
{7 315 (W) AnglePrint} 75 30 xyput
{(p) LabelAbovePoint} 50 60 xyput}

\par\noindent
Let $u = \LPROJ_W(p)$, $v = \LPROJ_{W^\perp}(p)$.  Draw and label
$W^\perp$, $u$, and $v$ in the picture.

\vspace{0.1in}

\problemno{8} {\bf [15 points]}
Let $A = \BRMAT{2&2&0\cr0&1&0\cr0&0&4}$.
\vspace{0.1in}
\par\noindent{\bf (a):}\ Explain how you can tell at a glance that $A$ is
invertible.
\par\noindent{\bf (b):}\ Compute $A^{-1}$.
\par\noindent{\bf (c):}\ For which numbers $r$ is the matrix $A-rI$ invertible?

\problemno{9} {\bf[10 points]}
Let $A$ be a square matrix such that $A A^T = I$.  What can one conclude about
$\det(A)$?  Justify your answer.

\problemno{10} {\bf[12 points]}
\par\noindent{\bf (a):}
What equation or equations must hold in order for the matrix
$\BRMAT{a&b\cr c&d}$ to be symmetric?
\par\noindent{\bf (b):}
Exhibit a basis for the vector space of all $2 \times 2$ symmetric matrices.

\problemno{11} {\bf [10 points]}
Let $V$ be the vector space of all continuous functions from $[0,1]$ to
$\R$.  Let $V$ have the inner product given by
$$\inn{f,g}\ =\ \int_0^1 f(t)g(t)dt.$$%
Let $h$ be the element of $V$ given by $h(t) = t+1$.  Compute $\norm{h}$.

\problemno{12} {\bf [12 points]}
A linear mapping \mp[[ f || \R^2 || \R^2 ]] is given by
counterclockwise rotation through the angle $\theta = 45^\circ$.

\widepost{230}{115}{
220 DottedAxes
nmv 30 cos 100 mul 30 sin 100 mul lineto stroke
nmv 75 cos 100 mul 75 sin 100 mul lineto stroke
newpath 30 cos 100 mul 30 sin 100 mul moveto 0 0 100 30 75 arc AltDotStroke
{Point 5 315 (p) AnglePrint} 30 cos 100 mul 30 sin 100 mul xyput
{Point 8 90 (f\(p\)) AnglePrint} 75 cos 100 mul 75 sin 100 mul xyput
newpath 30 cos 50 mul 30 sin 50 mul moveto 0 0 50 30 75 arc AltDotStroke
{6 225 {theta} MAnglePrint} 52.5 cos 50 mul 52.5 sin 50 mul xyput}

\par\noindent{\bf(a):}\ Write down the standard matrix of $f$.
[{\sc Hint.}\ There is more than one way to do this.  One way is to figure
out what $f$ does to the standard basis elements of $\R^2$.]
\vspace{1.5in}
\par\noindent{\bf(b):}\ If $p = \BRMAT{5\cr7}$, what is $f(p)$?

\problemno{13} {\bf[12 points]}
Let $M = \BRMAT{0&-1\cr1&0}$.
\vspace{0.05in}
\par\noindent{\bf(a):}\ Find the eigenvalues of $M$.
\vspace{2in}
\par\noindent{\bf(b):}\ Pick one eigenvalue $\lambda$ (your choice), and
find an eigenvector $x$ corresponding to $\lambda$.

{\Large
$$\lambda = \kern3in x =$$
}

\problemno{14} {\bf[18 points]}
Let $M = \BRMAT{1&2&3\cr2&3&1\cr3&1&2}$.

\par\noindent{\bf(a):}\ Let $x = \BRMAT{1\cr1\cr1}$, which is an eigenvector 
of $M$.  What is the corresponding eigenvalue?
\par\noindent{\bf(b):}\ If $n$ is a positive integer, what is $M^n x$?
\par\noindent{\bf(c):}\ Is $M$ diagonalizable?  You may use the fact that
the other two eigenvalues of $M$ are $\sqrt{3}$ and $-\sqrt{3}$.  Briefly 
justify your answer.