\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym, enumerate}
\usepackage[all]{xy}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\Char}{char}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\tr}{tr}

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}
\newcommand{\ds}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due April 3
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2015 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 7, pages 77--89. \\

\begin{enumerate}


% Lax Exercise 7.1 (p. 80)
\item Prove that $||x||=\max\{(x,y):y\in K^n\mbox{ with }||y||=1\}$. \\



\item Let $f$ and $g$ be continuous functions on the interval $[0,1]$.
  Prove the following inequalities.
  \begin{enumerate}
  \item $\displaystyle{\left(\int_0^1f(t)g(t)\,dt\right)^2\leq
    \int_0^1f(t)^2\,dt\int_0^1g(t)^2dt}$
  \item $\displaystyle{\left(\int_0^1(f(t)+g(t))^2\,dt\right)^{1/2}\leq
    \left(\int_0^1f(t)^2\,dt\right)^{1/2}+\left(\int_0^1g(t)^2dt\right)^{1/2}}$. \\
  \end{enumerate}



\item Use the Gram-Schmidt process to find an orthonormal basis for
  the subspace of $\R^4$ spanned by $y_1=(1,2,1,1)$, $y_2=(1,-1,0,2)$
  and $y_3=(2,0,1,1)$. \\



\item Let $X$ be the vector space of all continuous real-valued
  functions on $[0,1]$. Define an inner product on $X$ by
  \[
  (f,g)=\int_0^1 f(t)g(t)\,dt\,.
  \]
  Let $Y$ be the subspace of $X$ spanned by $f_0,f_1,f_2,f_3$, where
  $f_k(x)=x^k$. Find an orthonormal basis for $Y$. \\



% Lax Exercise 7.2 (p. 87)
\item Let $Y$ be a subspace of a Euclidean space $X$, and $P_Y\colon
  X\to X$ the orthogonal projection onto $Y$. Prove that $P_Y^*=P_Y$. \\



% Lax Exercise 7.5 (p. 89)
\item Show that a matrix $M$ is orthogonal iff its column vectors form
  an orthonormal set. \\



\item Let $X$ be an $n$-dimensional real Euclidean space, and $A\colon
  X\to X$ a linear map. Define the map $f\colon X\to X$ by
  $f(x,y)=x^TAy$. Give (with proof) necessary and suffcient conditions
  on $A$ for $f$ to be an inner product on $X$.


\end{enumerate}

\end{document}







  
