\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 5, due Wed Oct 19}

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

\vskip .05in

\ben

% 7 pts
\item Consider the matrix $A = \bma
1 &    3 &   -9 &    3\\
-1 &   -1  &  -3 &    1\\
2  &   3  &   0  &   0\ema$
\ben
\item Find a basis for Col $A$ :

\vspace{2.5in}

\item Find a basis for Nul $A$ :

\vspace{2.3in}


\item What is the dimension of the subspace spanned by the first
three columns of $A$?

\vspace{.8in}

\item For a {\em general} matrix $A$, do elementary row operations preserve
Col $A$ ? Nul $A$ ?
\een


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

% 7 pts
\item Let $H$ be
the subspace of vectors in $\RR^3$ whose entries sum to zero.

\ben
\item Each point in $H$ is a unique combination of the
linearly-independent vectors $\mbf{e}_1$, $\mbf{e}_2$, and $\mbf{e}_3$.
Explain whether or not this set of three vectors is a basis for $H$.

\vspace{2in}

\item What is dim $H$ ? Prove your answer.

\vspace{3in}

\item For a general vector space $V$ with basis
${\mathcal B}=\{\mbf{b}_1,\ldots,\mbf{b}_n\}$, prove that the
coordinate mapping $\mbf{x} \mapsto [\mbf{x}]_{\mathcal B}$
from $V$ to $\RR^n$ is one-to-one.



\een

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

% 6 pts
\item
Recall that $\{ 1,t,t^2\}$ form the standard basis for $\mathbb{P}_2$,
the vector space of polynomials of degree at most two.
In applications it is crucial to be able to
handle polynomials expressed about a new origin (eg, for Taylor series).
Let's shift the origin to the value $t=3$.

\vg

\ben
\item 
Prove that the set $\{1, \,t-3, \,(t-3)^2\}$ form a basis for $\mathbb{P}_2$.
[Hint: you may use that the coordinate mapping to $\RR^3$ is an isomorphism.]
% Work in $\RR^3$.

\vspace{3.5in}

\item
Say you are given the polynomial $a_0 + a_1t + a_2t^2$
but want to write it about the new origin,
$b_0 + b_1(t-3) + b_2(t-3)^2$.
Express each of the coefficients $b_0$, $b_1$, and $b_2$ in terms of
$a_0$, $a_1$, and $a_2$.

\vspace{3in}

\een

% invertible upper-triangular matrix

\een

\end{document}
