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\noindent Homework 8

\noindent Math 336, Winter `00

\noindent Due Friday, March 31


  
\vskip .2in

\noindent 
{\bf Problem 1:}  Compute the operator norm of the matrix
$$
\left(\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}\right).
$$
(Hint: a vector in $\mathbf{R}^2$ can be written in polar coordinates
$(x,y) = (r\cos\theta,r\sin\theta)$).
\bigskip

\noindent
{\bf Problem 2:} Show {\it using the definition of limit}
that the function $L:\mathbf{R^2} \to \mathbf{R^3}$
given by 
$$
L(x,y) = \left(\begin{array}{cc} 1 & -2 \\ e & 1 \\ 1 & \pi \end{array}\right)
         \left(\begin{array}{c} x \\ y \end{array}\right) + 
         \left(\begin{array}{c} 2 \\ 5 \\ 1 \end{array}\right)
$$
is continuous.
\bigskip

\bigskip
\noindent
{\bf Problem 3:} Show that the intersection of finitely many open sets
is open.  Give an example of a countable collection of open sets 
$U_j\subset\mathbf{R}^2$, 
$j\in\mathbf{N}$ whose intersection $\bigcap_{j=1}^\infty U_j$ is not open.
\bigskip

\bigskip
\noindent
{\bf Problem 4:} Let $f:\mathbf{R}^n\to\mathbf{R}^m$ be a continuous function
and let $U\subset\mathbf{R^m}$ be an open set.  Show that the
set
$$
f^{-1}(U) \eqdef \{\mathbf{x}\in\mathbf{R}^n: f(\mathbf{x}) \in U\}
$$
is open. 
\bigskip

\bigskip
\noindent
{\bf Problem 5:}
Let $f,g:\mathbf{R}^n\to\mathbf{R}^m$ be functions and $\mathbf{a}\in\R^n$,
be a point at which both 
$\lim_{\mathbf{x}\to\mathbf{a}} f(\mathbf{x})$ and
$\lim_{\mathbf{x}\to\mathbf{a}} g(\mathbf{x})$ 
exist.  Show that 
$
\lim_{\mathbf{x}\to\mathbf{a}} f(\mathbf{x}) + g(\mathbf{x})
$
exists and is equal to 
$
\lim_{\mathbf{x}\to\mathbf{a}} f(\mathbf{x}) + 
\lim_{\mathbf{x}\to\mathbf{a}} g(\mathbf{x}).
$
\bigskip

\bigskip
\noindent
{\bf Problem 6:}
\# 2 on page 244 of Wade's book---no proofs necessary.






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