\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}
\def\<{\langle}
\def\>{\rangle}

\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}\medskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\medskip}
\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 March 27
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2026 $|$ M.~Macauley}}


\begin{document}

$\;$\vspace{-2mm}



\begin{enumerate}

%%------------------------------------------------------------------------ 

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

%%------------------------------------------------------------------------ 
  
\item Let $X=\R^3$, and define the inner product by
  \[
  \<x,y\>=y^TAx=\begin{bmatrix}y_1&y_2&y_3\end{bmatrix}\begin{bmatrix}2 & -\sqrt{2} & 0 \\ -\sqrt{2} & 2 & -1 \\ 0 & -1 & 2\end{bmatrix}
  \vvv{x_1}{x_2}{x_3}.
  \]
  Find the norm of the three unit basis vectors $e_1$, $e_2$, and $e_3$, the angles between them, and the orthogonal complements of the lines that they span. \medskip
  
%%------------------------------------------------------------------------   
  
\item Given a linear map $A\colon X\to X$, define $f\colon X\to X$ by
  $f(x,y)=x^TAy$.
  \begin{enumerate}
  \item Write the inner product
    $f(x,y)=3x_1y_1-x_1y_2-x_2y_1+2x_2y_2-x_2y_3-x_3y_2+3x_3y_3$ as
    $f(x,y)=x^TAy$.
  \item Find an orthonormal basis $v_1,v_2,v_3$ of $\R^3$ so that with
    respect to this basis, $f(z,w)=z^TDw$ for some diagonal matrix
    $D$.
  \item Write a formula for $f(z,w)$ like in Part~(b), but with
    respect to this new basis. \medskip
  \end{enumerate}
  
%%------------------------------------------------------------------------ 

\item Let $Y,Z$ be subspaces of an inner product space $X$.
  \begin{enumerate}
  \item Show that $Y\subseteq Y^{\perp\perp}$, with equality holding
    if $\dim X<\infty$.
  \item Give an example of an infinite dimensional space where equality
    does not hold. \medskip
  \end{enumerate}
  
%%------------------------------------------------------------------------ 
  
  \item Let $Y,Z$ be subspaces of an inner product space. Show that
    $(Y+Z)^\perp=Y^\perp\cap Z^\perp$. \medskip
    
%%------------------------------------------------------------------------ 

\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 $y_3=(2,0,1,1)$. Then write the vector $v=(4,1,2,4)$ in this basis. \medskip

%%------------------------------------------------------------------------ 

\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
  $f_k(x)=x^k$.
  \begin{enumerate}
  \item Use the Gram-Schmidt process to construct an orthonormal basis
    for $Y$.
  \item Write $f(x)=2x^3-x^2+4$ using your basis from Part~(a).
  \end{enumerate}
    
\end{enumerate}

\end{document}

  

