\documentclass{math25}
% By Tom Braden
% Minor editing by Nils Barth

\title{Math 25a -- Honors Advanced Calculus and Linear Algebra\\
\rm Final Examination, January 23, 1998}

\begin{document}
\raggedbottom

\noindent {\bf Name:}

The questions are listed first and then repeated, one to a page.
Answer each one on its own page.  You can continue an answer on
the back of the page and on the blank sheets at the end of the exam.
If you need to hand in extra sheets, {\bf write your name on
each sheet} that you hand in.  Try to make your explanations
clear and complete.

This exam is fairly long, so budget your time carefully.  All
problems are worth the same amount.  The later problems are
(I think) somewhat harder.  Except where otherwise stated, you
can use any theorem from the course (be sure to state it carefully!).
You can assume an earlier part of a question if it helps you do a
later part.

Note: $\R^n$ always carries the standard inner product and its
associated norm.

Good Luck!

\begin{enumerate}
\item Let $X$ be a metric space, and let $K\subset X$ be a compact subset.
\begin{enumerate}
\item Give a precise definition of the phrase ``$K$ is compact''.
\item Show directly from the definition that $K$ is
bounded.
\item Now suppose $X = \R$, and $C\subset \R$ is a closed subset.
Show that the set $K + C = \set{x+y \mid x \in K,\, y\in C}$ is
closed.
\end{enumerate}

\item Let the linear transformation $T\colon \R^4 \to \R^3$ be
given by the matrix
\[\left(\begin{array}{rrrr}
2 & 1 & 0 & 1 \\
-2 & 2 & 3 & 2 \\
1 & 2 & \frac{3}{2} & 2
\end{array}\right).\]
\begin{enumerate} \item Find an {\bf orthonormal} basis for each of
$\ker T$ and $\im T$.
\item Calculate the $3\times 3$ matrix for the orthogonal projection
of $\R^3$ onto $\im T$.
\end{enumerate}

\item Find, with proof, the points in the unit ball $B = \set{(x, y) \in \R^2
\mid x^2 + y^2 \le 1}$ where the function $f(x,y) = x^2y^3$ takes on
its maximum and minimum values.

\item Let $T\colon \R^n \to \R^n$ be a linear transformation.
\begin{enumerate}
\item Show that there exists a unique linear transformation $S$ so that
$\langle Tx, y\rangle = \langle x, Sy\rangle$ for all $x, y \in \R^n$.
\item Show that if $T$ is invertible, then $S$ is also.  (More challenging:
show that $S$ and $T$ have the same rank).
\end{enumerate}

\item \begin{enumerate}
\item Carefully state Taylor's theorem for functions $f\colon \R \to \R$.
\item Suppose $f\colon \R\to \R$ is a $\cC^4$ function, that
$f'(0) = f''(0) = f'''(0) = 0$, and $f''''(0) = 1$.  Prove that
$f$ has a local minimum at $0$.
\end{enumerate}


\item Let $f\colon V\to W$ be a function between normed vector spaces.
\begin{enumerate}
\item Suppose that $f$ is differentiable at ${\vec v}_0 \in V$, and
the derivative $D_{{\vec v}_0}f$ is a bounded linear transformation
(i.e.\/ there is an $M$ so that for all $\vec v\in V$,
$\lnorm D_{{\vec v}_0}f(\vec v)\rnorm \le M\lnorm \vv\rnorm)$.  Show that
$f$ is continuous at $\vv_0$.
\item Now suppose that $f$ is differentiable at all points of $V$, and
and the derivative is ``uniformly bounded'', \ie we can find a
single number $M$ which works for all $\vv_0$.  Show that $f$ is
uniformly continuous.
\end{enumerate}

\item
\begin{enumerate}
\item Let $W \subset \R^n$ be a subspace, and take a vector $\vv \in \R^n$
which is not in $W^\bot$.
Let $\vec w_0$ be the unit vector in $W$ which has the
largest possible inner product with $\vv$.  Show that ${\vec w}_0$
 is a multiple of $\mathop{\rm Pr}_W \vv$, the orthogonal projection
 of $\vv$ onto $W$.
\item Let $S$ be the sphere in $\R^3$ of radius $3$ and center $(0,0,0)$.
A particle travels with constant speed $1$
along a differentiable path $\alpha\colon \R \to S$.
At time $t = 0$ it is at the point $(-1, 2, 2)$.  If it is moving so
as to increase the function $f(x, y, z) = x^2+ xy + yz$ as much as possible,
calculate the velocity vector $\alpha'(0)$.
\end{enumerate}

\item Let $T_1, T_2 \colon \R^n \to \R^m$ be two linear transformations,
with ranks $k_1$ and $k_2$.
\begin{enumerate}
\item Show that $\rank(T_1 + T_2) \le k_1 + k_2$.
\item Suppose that $k_1 \le k_2$.  Show that $\rank(T_1 + T_2) \ge k_2 - k_1$.
\item Give examples where $n = m = 2$, $k_1 = k_2 = 1$, and $\rank(T_1 + T_2)$
is $0$, $1$, and $2$.
\end{enumerate}

\item Let $X$ be a metric space.
Recall that we say a sequence of functions $f_n\colon X \to \R$
converges uniformly to a function $f$ if for any $\epsilon >0$ there
exists an $N>0$ so that if $n \ge N$, then for all $x\in X$ we have
$|f_n(x) - f(x)| \le \epsilon$.
\begin{enumerate}
\item Suppose $\seq{f_n}$ is a sequence of differentiable
functions defined on $(-1,1)$, $f'_n \to f'$ uniformly, and $f_n(0) = f(0)$
for all $n$.  Show that $f_n \to f$ uniformly.
\item Show that if the functions were defined on all of $\R$, we would
still get $f_n(x_0) \to f(x_0)$ for all $x_0$, but the convergence
might not be uniform.
\end{enumerate}


\end{enumerate}
\end{document}
