\documentclass[11pt]{amsart}
\pagestyle{empty}

\addtolength{\hoffset}{-48pt}
\addtolength{\textwidth}{96pt}
\addtolength{\voffset}{-32pt}
\addtolength{\textheight}{45pt}

\usepackage{graphicx}

\newcommand{\RR}{{\mathbb R}}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}
\newcommand\ben{\begin{enumerate}}
\newcommand\een{\end{enumerate}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand\vg{\vspace{2ex}}
\newcommand{\bc}{\begin{center}}
\newcommand{\ec}{\end{center}}
\newcommand{\ie}{{\it i.e.\ }}
\newcommand{\eg}{{\it e.g.\ }}
\newcommand{\mbf}[1]{{\mathbf #1}}
\newcommand{\bma}{\left[\begin{matrix}}
\newcommand{\ema}{\end{matrix}\right]}   % note alex likes \em as it is!

\begin{document}

Your name:
\vskip 12pt

Instructor (please circle): 
\hspace{1in}
Alex Barnett
\hskip .7in
Naomi Tanabe
\vskip 12pt

{\bf \large Math 22 Fall 2016, Homework 8, due Wed Nov 9}

{\em Please show your work. No credit is given for solutions without work or justification.}

\vskip .05in

\ben

% 7 pts
\item
Let 
$\mbf{v}_1 = \bma1\\1\\1\\-1\ema$,
$\mbf{v}_2 = \bma1\\3\\1\\5\ema$, and
$\mbf{v}_3 = \bma1\\0\\-1\\0\ema$
and let $W =$ Span $\{\mbf{v}_1,\mbf{v}_2,\mbf{v}_3\}$.
In this question do not use row reduction.
\ben
\item
Use the fact that each vector in the set $\{\mbf{v}_j\}$ is orthogonal to the other two
to write $\mbf{y} = \bma 0\\0\\-4\\8\ema$ as a linear combination of
the three vectors.

\vspace{3in}

%\item Write an orthonormal basis for $W$

%\vspace{2in}

\item Write $W^\perp$ as the null space of a certain matrix.

\vspace{1.5in}

\item What is dim $W^\perp$ ? (Explain)

\een

\newpage %%%%%%%%%%%%%%%%

% 7 pts
\item 
\ben
\item Compute the point in Span $\left\{\bma -4\\-1\\1\ema, \bma 0\\1\\1\ema\right\}$ which has the minimum distance from the point $\bma 7\\2\\3\ema$.

\vspace{3in}

\item What is the distance in part (a) ?

\vspace{1.5in}

\item Let $A$ be any $n\times n$ matrix whose columns are an orthonormal basis
for $\RR^n$. Prove the amazing result that its rows must also be an orthonormal basis for $\RR^n$.
(As always with proofs, if you use previous theorems, explain how.)

\een

\newpage %%%%%%%%%%%%%%%%

% 6 pts
\item
Consider the matrix $A = \bma1&1&1\\0&1&1\\0&1&0\\0&0&1\ema$.
\ben
\item Use the Gram-Schmidt process to compute an orthogonal basis for Col $A$.

\vspace{4in}

\item Use the above to compute the factors $Q$ and $R$ in the
$QR$ factorization of $A$.

\een
\een

\end{document}
