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% Homework for the course "Math 554: Linear Analysis",
% Autumn quarter, 2008, Anne Greenbaum.
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\noindent
{\scriptsize Math 554, Autumn 2008} \hfill
\begin{center}
\large
Assignment 1. Due Friday, Oct. 3.
\normalsize
\end{center}
\begin{tabular}{ll}
Reading: & Read through p. 12 in Course Notes. \\
& Horn and Johnson, Chapter 0. \\
& Review linear algebra from Halmos, Nering, or your choice of \\
& similar linear algebra text.
\end{tabular}
\vspace{.3in}
% The questions!
\begin{enumerate}
\item
Let $W_1$ and $W_2$ be finite dimensional subspaces of a vector space $V$
and define
\[
W_1 + W_2 = \{ w_1 + w_2 :~w_1 \in W_1 , ~w_2 \in W_2 \} .
\]
\begin{description}
\item[(a)]
Show that
\[
\mbox{dim}( W_1 + W_2 ) + \mbox{dim}( W_1 \cap W_2 ) =
\mbox{dim}( W_1 ) + \mbox{dim}( W_2 ) .
\]
\item[(b)]
Show that the following conditions are equivalent:
\begin{description}
\item[(i)]
$W_1 \cap W_2 = \{ 0 \}$.
\item[(ii)]
Each vector $v \in W_1 + W_2$ can be written uniquely as $v = w_1 + w_2$,
where $w_1 \in W_1$ and $w_2 \in W_2$.
\item[(iii)]
If $w_1 \in W_1$ and $w_2 \in W_2$ and $w_1 + w_2 = 0$, then
$w_1 = w_2 = 0$.
\end{description}
\end{description}
\item
Let $V = \{ u \in C^4 ( {\bf R} ) :~u'''' + u'' = 0 \}$ and let
$W \subset V$ be the subspace of functions satisfying in addition
$u( \pi ) = 0$.
\begin{description}
\item[(a)]
Find bases for $V$ and $W$.
\item[(b)]
Show that the map $u \rightarrow u'$ defines a linear transformation
from $V$ to $V$. Hence we can restrict the domain to $W$ to obtain
a linear transformation $L: W \rightarrow V$.
\item[(c)]
Find the matrix of $L$ with respect to your bases in part (a).
\end{description}
\item
Show that if $0 < p < q \leq \infty$, then ${\ell}^p \subset {\ell}^q$
but ${\ell}^p \neq {\ell}^q$.
\item
Let ${\cal P}_n$ denote the vector space of polynomials of degree
less than or equal to $n$. Let $x_0 , x_1 , \ldots x_n$ be $n+1$
distinct points in ${\bf R}$ and define a linear functional
$f_k \in {\cal P}_n^{*}$ by
\[
f_k ( p ) = p( x_k ) ,
\]
for $p \in {\cal P}_n$. Let $\{ e_0 , e_1 , \ldots , e_n \}$
be the basis for ${\cal P}_n^{*}$ dual to the basis
$\{ 1, x, \ldots , x^n \}$ for ${\cal P}_n$.
\begin{description}
\item[(a)]
Express $f_k$ as a linear combination of $e_0 , \ldots , e_n$.
\item[(b)]
Show that $\{ f_0 , \ldots , f_n \}$ is a basis for ${\cal P}_n^{*}$.
(Hint: Vandermonde matrix)
\item[(c)]
Write down the basis of ${\cal P}_n$ to which $\{ f_0 , \ldots , f_n \}$
is dual.
\end{description}
\item
Let $[a,b] \subset {\bf R}$ be a closed bounded interval, and let
$\Omega_n = \{ x_0 , \ldots , x_n \}$ be a fixed partition of $[a,b]$;
i.e., $a = x_0 < x_1 < \ldots < x_{n-1} < x_n = b$. To approximate
the solution to boundary value problems in differential equations,
one often uses {\em piecewise polynomials}: functions that are
equal to polynomials of a specified degree in each subinterval
$[ x_i , x_{i+1} ]$ and usually have some continuity properties
throughout the entire interval $[a,b]$.
\begin{description}
\item[(a)]
Show that the set of functions
\[
\varphi_i (x) = \left\{ \begin{array}{cl}
\frac{x - x_{i-1}}{x_i - x_{i-1}} , & x \in [ x_{i-1} , x_i ] \\
\frac{x_{i+1} - x}{x_{i+1} - x_i} , & x \in [ x_i , x_{i+1} ] \\
0 & \mbox{otherwise} \end{array} \right. ,~~~i=1, \ldots , n-1 ,
\]
forms a basis for the set of continuous piecewise linear functions
$\varphi$ on $[a,b]$ satisfying $\varphi (a) = \varphi (b) = 0$.
[Note that the coordinates of $\varphi$ with respect to this basis
are the values of $\varphi$ at the interior nodes:
$\varphi (x ) = \sum_{i=1}^{n-1} \varphi ( x_i ) \varphi_i (x)$.]
\item[(b)]
Write down a basis for the set of $C^1$ piecewise cubic functions
$\psi$ on $[a,b]$ satisfying $\psi (a) = \psi' (a) = \psi (b) =
\psi' (b) = 0$. [Hint: Consider two types of basis functions:
piecewise cubics $\psi_i$ that are one at node $i$ and zero
at all other nodes and whose first derivatives are zero at all
of the nodes; and piecewise cubics $\omega_i$ that are zero at all
of the nodes but have first derivative equal to one at node $i$ and
zero at all other nodes. These functions are called {\em Hermite cubics}.]
\end{description}
\end{enumerate}
\end{document}