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\par\noindent{\Large\bf Math 314/814 \kern20pt -- \kern20pt Final Exam\kern20pt 
-- \kern20pt May 2, 1995}
\vspace{0.30in}

\problemno{1} {\bf[16 points]}

\par\noindent{\bf(a):}\ Compute the determinant of the following matrix:
$\BRMAT{2&1&4&5&6\cr5&6&6&1&7\cr0&0&x&4&8\cr0&0&0&5&\pi\cr0&0&0&0&-1}$.

\par\noindent{\bf(b):}\ A $3 \times 3$ matrix $A$ satisfies the equation
$A A^T = 4I$.  Determine all possible values for the determinant of $A$.
(Show your work carefully.)

\problemno{2} {\bf[10 points]}
Let $V$ be an inner product space, and let $x,y,z,w \in V$.  Given that
$\inn{x,x} = 1$, $\inn{x,y} = 2$, $\inn{x,z} = 3$, $\inn{x,w} = 4$,
$\inn{y,y} = 5$, $\inn{y,z} = 6$, $\inn{y,w} = 7$, $\inn{z,z} = 8$,
$\inn{z,w} = 9$, and $\inn{w,w} = 10$, compute $\inn{x+2y, z-w}$.

\problemno{3} {\bf[18 points]}
Determine in each of the following cases if the given set $W$ is a subspace of 
the given vector space $V$.  If so, prove it.  If not, show that some property
of subspaces does not hold.

\vspace{0.1in}

\par\noindent{\bf(a):}\ $V = $ the vector space of all $2 \times 2$ real
matrices, $W = $ the subset consisting of those matrices having 
$\rank$ $\leq 1$.

\par\noindent{\bf(b):}\ $V = \R^3$, $W = $ all elements of $\R^3$ which
are orthogonal to the vector $x = (1,2,3)$.  [Use the dot product as inner 
product on $\R^3$.]

\problemno{4} {\bf[10 points]}
Let $A$, $B$, $C$, and $X$ be invertible $2 \times 2$ matrices which
satisfy the equation $AXC = B^{-1}A$.  Solve for $X$ in terms of 
$A$, $B$, $C$ and their inverses.

\problemno{5} {\bf[20 points]}
In each part exhibit (if possible) a specific matrix of the indicated 
type; write {\it impossible\/} otherwise.  Your matrix should not have
variables in it.

\vspace{0.1in}

\par\noindent{\bf(a):}\ a $2 \times 2$ invertible matrix having
     determinant $-1$

\par\noindent{\bf(b):}\ a $2 \times 3$ matrix having rank $3$

\par\noindent{\bf(c):}\ a $2 \times 2$ invertible matrix having
     eigenvalues $0$ and $2$

\par\noindent{\bf(d):}\ a $4 \times 3$ matrix whose columns form
     an orthonormal set.  (Use the dot product as inner product.)

\problemno{6} {\bf[10 points]}
Let $A = \BRMAT{0&1\cr1&1}$, $B = \BRMAT{1&3\cr2&4}$.  Find all 
$2 \times 2$ matrices $X$ such that $AX = B$.

\problemno{7} {\bf[14 points]}
Let $V$ be the vector space of all polynomials (with real coefficients),
and define an inner product on $V$ by
$$\inn{f,g} = \int_0^1 f(t)g(t)dt.$$%
Denote by $W$ the subspace of $V$ consisting of all polynomials having
the form $a+bt$, where $a,b$ are real numbers.  In other words, $W$ consists
of all polynomials in $t$ of degree $\leq 1$.

\vspace{0.1in}

\par\noindent{\bf(a):}\ Exhibit a basis for $W$.

\par\noindent{\bf(b):}\ Exhibit an orthonormal basis for $W$.

\problemno{8} {\bf[8 points]}
Suppose one has a system of $5$ linear equations in $5$ variables,
whose coefficient matrix is {\it not\/} invertible.  Check all outcomes
listed below which are possible, given this information:

\vspace{0.1in}

\par\noindent{\bf(a):}\ the system has no solutions\ \ \bocks
\vspace{0.1in}
\par\noindent{\bf(b):}\ the system has exactly one solution\ \ \bocks
\vspace{0.1in}
\par\noindent{\bf(c):}\ the system has exactly two solutions\ \ \bocks
\vspace{0.1in}
\par\noindent{\bf(d):}\ the system has infinitely many solutions
\ \ \bocks\kern3pt.

\problemno{9} {\bf[18 points]}
Exhibit a basis for each of the following subspaces of $\R^4$:

\vspace{0.1in}

\par\noindent{\bf(a):}\ $\span\left\{ \BRMAT{1\cr0\cr1\cr0},
   \BRMAT{2\cr0\cr3\cr0}, \BRMAT{0\cr0\cr5\cr0}, \BRMAT{3\cr0\cr2\cr0}
   \right\}$

\par\noindent{\bf(b):}\ $\NullSpace(A)$, where 
$A = \BRMAT{1&0&3&4\cr0&1&0&0}$

\problemno{10} {\bf[14 points]}
\par\noindent{\bf(a):}\ Complete the following definition.  {\bf An 
$n \times n$ matrix $A$ is {\it diagonalizable\/} if}
\vspace{0.15in}
\par\noindent\blank6
\vspace{0.15in}
\par\noindent\blank6

\vspace{0.2in}

\par\noindent{\bf(b):}\ Let $A$ be a diagonalizable matrix.  Prove that
$A^2$ is diagonalizable.

\problemno{11} {\bf[21 points]}
In each case determine if the given elements are linearly
dependent or independent.  If they are dependent, explain how you
know this.

\vspace{0.1in}

\par\noindent{\bf(a):}\ $\BRMAT{1\cr0\cr0}$, $\BRMAT{0\cr1\cr0}$,
    $\BRMAT{0\cr1\cr1}$

\par\noindent{\bf(b):}\ $\BRMAT{1\cr2\cr3\cr4}$, $\BRMAT{5\cr0\cr6\cr0}$,
   $\BRMAT{1\cr2\cr0\cr0}$, $\BRMAT{3\cr0\cr0\cr0}$, $\BRMAT{4\cr3\cr2\cr1}$

\par\noindent{\bf(c):}\ $\BRMAT{1\cr0\cr0\cr0}$, $\BRMAT{5\cr2\cr4\cr3}$,
   $\BRMAT{0\cr0\cr1\cr0}$, $\BRMAT{4\cr0\cr7\cr0}$

\problemno{12} {\bf[9 points]}
Let $A = \BRMAT{1&i\cr-i&3}$, $B = \BRMAT{2&2+i\cr0&4}$.  Compute $AB$.

\problemno{13} {\bf[14 points total]}
Let $A = \BRMAT{1&3\cr0&2}$.

\vspace{0.1in}

\par\noindent{\bf(a):}\ {\bf[4 points]}\ What are the eigenvalues of $A$?

\par\noindent{\bf(b):}\ {\bf[10 points]}\ Exhibit a $2 \times 2$
invertible matrix $P$ such that $P^{-1}AP = \BRMAT{1&0\cr0&2}$.

\problemno{14} {\bf[18 points]}
Let $\R^3$ have the standard inner product, i.e.\ the dot product.  Let
$W$ be the subspace of $\R^3$ consisting of all vectors of the form
$(t,-t,t)$, where $t \in \R$.

\vspace{0.1in}

\par\noindent{\bf(a):}\ Exhibit a basis for $W$.

\par\noindent{\bf(b):}\ Exhibit a basis for $W^\perp$.

\par\noindent{\bf(c):}\ Compute the projection of $(1,2,3)$ onto $W$.