\documentstyle[12pt]{article}

\input amssym.def
\input amssym
\newcommand{\reals}{{\Bbb R}}
\newcommand{\by }{\times}
\newcounter{xprobno}
\newcommand{\xprob}[1]{\addtocounter{xprobno}{1}
   \par\medskip\noindent\thexprobno . (#1 points) }
\newcommand{\true }{\addtocounter{xprobno}{1}
   \par\medskip\noindent\thexprobno . }
\newcommand{\false }{\addtocounter{xprobno}{1}
   \par\medskip\noindent\thexprobno . Circle one: True or False}
\newcommand{\norm}[1]{||#1||}
\newcommand{\complex }{\Bbb C}
\newcommand{\field }{\Bbb F}

\begin{document}

\begin{flushright}
  A. Miller\\
  M340 Final Exam\\
  May 14, 1991
\end{flushright}


{\bf There is a total of 200 points.
You can pick up your graded exam in VanVleck 403.
Your grade will be posted on the door. }


\xprob{10} Find the inverse of $A$ or show that the inverse does not exist.
$$A=
\left[
\begin{array}{rrrr}
  0 & 0 & 0 & -1  \\
  0 & 0 & 1 & 0  \\
  0 & -1 & 0 & 0  \\
  1 & 0 & 0 & 0  \\
\end{array}
  \right]
$$

\xprob{20} Suppose that
$M$ and $N$ are invertible matrices such that
$MAN=B$.

a. If $H$ is a basis matrix for ${\cal C}(B)$, then
what matrix is a basis matrix for ${\cal C}(A)$?

b. If $K$ is a basis matrix for ${\cal N}(A)$, then
what matrix is a basis matrix for ${\cal N}(B)$?

c. If $R$ is a right inverse of $A$, then what matrix is
a right inverse of $B$?

d. If $L$ is a left inverse of $B$, then what matrix is
a left inverse of $A$?

\xprob{20} Find a diagonal matrix $D$ and invertible matrix $P$ such
that $A=PDP^{-1}$ where
$$
A=
\left[
\begin{array}{rr}
 4 & -1 \\
 2 & 1
\end{array}
\right]
$$


\xprob{10} $\langle\left[
\begin{array}{c}
  1+i \\ 1
\end{array}
\right] ,
\left[
\begin{array}{ccc}
 1+i\\ 3
\end{array}
  \right]
\rangle$=

\xprob{10} $|2+3i|$=

\xprob{10} $\norm{\left[
\begin{array}{ccc}
 i\\1
\end{array}
  \right]}$=


\newpage

\xprob{15} Let $T:V\rightarrow W$ be a mapping.

a. Define $T$ is one-to-one.

b. Prove that if T is one-to-one, then for every $X,Y\subseteq V$
$$T(X)\cap T(Y)\subseteq T(X\cap Y).$$

\xprob{15} Prove:  For all $U,V\in\reals^{n\by 1}$
$$|\langle U,V\rangle|\leq \norm{U}\;\norm{V}$$

\xprob{15} Prove that if $A\in\complex^{n\by n}$ is Hermetian
(ie. $\overline{A}^T=A$), then the eigenvalues of $A$ are real.

%%%%%%%%%%%%% true and false %%%%%%%%%%%%%%%%%

\begin{center}
  True or false, enter your answers on the answer sheet.
\end{center}

\true If a real matrix is both unitary and Hermetian, then
it must be the identity matrix.

\true If $U$ and $V$ are $n\by n$ uppertriangular matrices
then so is $UV$.

\true If $D$ is a diagonal $n\by n$ matrix, then so is $cD^{n+1}$
where $c$ is a scalar.

\true If $A,B,C$ and $D$ are $n\by n$ matrices, then $(AB)(CD)=A(BC)D$.

\true If $A$ and $B$ are $n\by n$ matrices and
$c$ is a scalar, then $A(cB)=B(cA)$.

\true If $A,B$, and $C$ are $n\by n$ matrices and $A+C=C+B$, then
$A=B$.

\true If $A$ is an $n\by n$ invertible matrix and $B$ is $n\by 1$, then
the matrix equation $AX=B$ has a unique solution.

\true If $U$ and $V$ are unitary $n\by n$ matrices, then
$UV$ is a unitary matrix.

\true An $n\by n$ matrix $U$ is unitary iff
for every $X$ which is $n\by 1$ we have $\norm{UX}=\norm{X}$.

\true If $A$ and $B$ are Hermetian $n\by n$ matrices, then
$A+B$ is Hermetian.

\true If $A,B$, and $C$ are $n\by n$ matrices and $AC=BC$, then
either $A=B$ or $C$ is the zero matrix.

\true It is possible to find $5\by 1$ vectors $U$ and $V$ such
that $\norm{U}$=2, $\norm{V}=3$ and $\norm{U+V}=4$.

\true It is possible to find $17\by 1$ vectors $U$ and $V$ such
that $\norm{U}$=4, $\norm{V}=2$ and $\norm{U+V}=7$.

\true It is possible to find $3\by 1$ vectors $U$ and
$V$ such that $\langle U,V\rangle=5$ and $\norm{U}=\norm{V}=2$.

\true For any $n$ there exists an $n\by n$ elementary matrix E such
that $EA=A^T$ for any $n\by n$ matrix $A$.

\true For any $m\by n$ matrix $A$ it is possible to find
an invertible $m\by m$ matrix $M$ such that $MA$ is in row echelon
form.

\true The sum of two $n\by n$ invertible matrices is invertible.

\true If $A$ and $B$ are similar matrices, then $trace(A)=trace(B)$.

\true If $A$ and $B$ are similar, then $det(A)=det(B)$.

\true If two matrices
$A$ and $B$ have the same characteristic polynomial, then they are
similar.

\true If two matrices
$A$ and $B$ are similar, then they have the same characteristic polynomial.

\true Suppose that $A\in \field^{n\by m}, B\in \field^{m\by p},
C\in \field^{p\by r}$.  Then  $(AB)C=A(BC)$.

\true
If $W_1$ and $W_2$ are subspaces of $\field^{n\by 1}$, then define
$$W_1\bigoplus W_2=\{w_1+w_2:w_1\in W_1,w_2\in W_2\}.$$
Then $W_1\bigoplus W_2$ is a subspace of $V$.

\true Suppose that $u_1,u_2$ are orthogonal unit vectors in
$\reals^{n\by 1}$ and $c_1,c_2\in\reals$, then
$\norm{c_1u_1+c_2u_2}=\sqrt{c_1^2+c_2^2}$.

\true The inverse of a Hermetian matrix is Hermetian.

\newpage

\begin{center}
  Answer sheet for true and false.\\
  Each one is worth 3 points.
\end{center}

\setcounter{xprobno}{9}

\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false
\false

\end{document}