\documentclass{amsart}

\usepackage{amsmath}
\usepackage{amssymb}

\pagestyle{empty}
\textwidth 7.2in
\textheight 10in
\oddsidemargin -0.1in
\evensidemargin -0.2in
\topmargin -0.3in



\input{macros}

\begin{document}
\noindent
\Large Math 554\hfill Autumn 2014

\noindent
Homework Set 2\hfill Due Friday October 10
\bigskip

\begin{flushleft}
\begin{tabular}{lp{5.5in}}
\kern-.1in{\bf Reading} &  Continue your review of linear algebra, including
determinants in H-J.
\end{tabular}
\end{flushleft}

\begin{enumerate}
\item 
Define subspaces $W_1, W_2\subset C[0,1]$ by
$W_1=\{f:\int^1_0f(x)dx=0\}$ and $W_2=\{f:f(x)=c\mbox{ for some
}c\in\C\mbox{ and all }x\in[0,1]\}$. Show that $C[0,1]=W_1\oplus W_2$, and
derive explicit formulas for the projection operators $P_1$, $P_2$ onto
$W_1$, $W_2$.
\item 
Let $V,W$ be finite-dimensional vector spaces of dimension $n,m$,
respectively. Let $L\in\B(V,W)$, and suppose $\rank(L)=1$. Show that for any
choice of bases for $V$ and $W$, the matrix of $L$ is of the form $ab^T$
where $a=\left(\begin{array}{c}a_1\\ \vdots\\ a_m\end{array}\right)$,
$b=\left(\begin{array}{c}b_1\\ \vdots\\ b_n\end{array}\right)$. (This matrix
is called the \index{outer product}\index{product!outer}\emph{outer product}
of the vectors $a$ and $b$.)
\item
Let $\P_n$ be the space of polynomials of degree $n$ or less, and
consider the differential operator $D$ as a mapping from
$\P_n$ to $\P_n$ ($D\,:\,\P_n\mapsto \P_n$).
\begin{enumerate}
\item
Show that $D$ is nilpotent.
\item
Provide at least two different bases in which the matrix representation
for $D$ in these bases is a 
direct product of shift operators.
\end{enumerate}
\item 
(In this problem, take $\F=\R$.) Let $A\in\R^{n\times n}$. One can
regard the entries $a_{ij}$ as independent variables, and $\det A$ as a
function of these $n^2$ variables.
\begin{enumerate}
\item Show that $\frac{\p}{\p a_{ij}}(\det A)=\widehat A_{ij}$, where
$\widehat A_{ij}$ is the $(i,j)$ cofactor of $A$, i.e., $\widehat
A_{ij}=(-1)^{i+j}\det(A[i|j])$, where $A[i|j]$ is the $(n-1)\times(n-1)$
submatrix of $A$ obtained by removing the $i^{\mathrm{th}}$ row and
$j^{\mathrm{th}}$ column of $A$.
\item Suppose $A(t)$ is an $\R^{n\times n}$-valued function of $t\in\R$
whose entries $a_{ij}(t)$ are differentiable functions of $t$. Show that
$\frac{d}{dt}(\det A(t))=\sum^n_{i=1}\sum^n_{j=1}\widehat
A_{ij}(t)\frac{d}{dt}(a_{ij}(t))$.
\item Suppose in addition $A(0)=I$. Show that $\frac{d}{dt}(\det
A(t))\Bigg|_{t=0}=\tr\left(\frac{dA}{dt}(0)\right)$.
\item Suppose $A(t)$ is as in part (b), and suppose $\det A(t)\ne 0$ for all
$t\in\R$. Show $\frac{d}{dt}\log(\det
A(t))=\sum^n_{i=1}\sum^n_{j=1}(A^{-1})_{ji}\frac{d}{dt}(a_{ij}(t))$, where
$(A^{-1})_{ji}$ is the $ji^{\mathrm{th}}$ entry of $A(t)^{-1}$.
\end{enumerate}
\item This problem continues and extends problem 2 on Problem Set 1.
\begin{enumerate}
\item (extension of 2(b) on P.S.1) Let $\{a_0,a_1,\ldots,a_n\}$ be distinct
real numbers. Show that the linear functionals $f_0,\ldots, f_n$ on $\P_n$
defined by $f_k(p)=p(a_k)$ form a basis of $\P^\ast_n$.
\item Define polynomials $\ell_0,\ldots,\ell_n\in\P_n$ by
$\ell_k(x)=\prod_{0\le i\le n, i\ne k} \left(\frac{x-a_i}{a_k-a_i}\right)$.
Show that $\ell_0,\ldots,\ell_n$ are linearly independent, and thus form a
basis of $\P_n$. (Hint: first show $\ell_k(a_i)=\d_{ik}$.) The $\ell_k$'s
are called the \index{Lagrange
polynomials}\index{polynomials!Lagrange}\emph{Lagrange polynomials}
(associated with the set $\{a_0,\ldots, a_n\}$).
\item Show that $\{f_0,\ldots, f_n\}$ in part (a) form the basis of
$\P^\ast_n$ which is dual to the basis $\{\ell_0,\ldots,\ell_n\}$ of $\P_n$.
\item Show that if $c_0,\ldots, c_n\in\C$, then
$p(x)=\sum^n_{k=0}c_k\ell_k(x)$ is the unique element of $\P_n$ satisfying
the interpolation conditions $p(a_k)=c_k$, $0\le k\le n$. This result is
called the Lagrange interpolation formula. It provides an explicit formula
for a polynomial $p$ of degree at most $n$ having prescribed values
$c_0,\ldots, c_n$ at the prescribed distinct points $a_0,\ldots, a_n$.
\end{enumerate}
\item Formulate and prove a precise statement of the following fact:
\begin{enumerate}
\item[]If $\{v_1,\ldots, v_k\}$ is a linearly independent set of vectors in
$\R^n$ (clearly $k\le n$), and each of $v_1,\ldots, v_k$ is perturbed
slightly, then the resulting set of vectors is also linearly independent.
\end{enumerate}
\end{enumerate}


\end{document}
