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\begin{document}

\title{Chapter Check for Chapter 2}
\maketitle

1. Let $P_{k}\left(  \mathbb{R}\right)  $ denote the set of polynomials in one
variable of degree at most $k$ with coefficients in $\mathbb{R}$. Let
$Y=\left\{  y_{0},\ldots,y_{k}\right\}  $ and recall that $\mathcal{F}\left(
Y,\mathbb{R}\right)  $ is the vector space of real-valued functions from $Y$
to $\mathbb{R}$.

\bigskip

\qquad a. Explain why $P_{k}\left(  \mathbb{R}\right)  $ and $\mathcal{F}%
\left(  Y,\mathbb{R}\right)  $ are isomorphic without finding an isomorphism
(i.e., use a theorem).

\bigskip

\qquad b. Find an explicit isomorphism and show it is an isomorphism.

\bigskip

2. Consider the vector spaces $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ with
ordered bases $\beta=\left\{  \left(
\begin{array}
[c]{c}%
1\\
1
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
2\\
0
\end{array}
\right)  \right\}  $ and $\gamma=\left\{  \left(
\begin{array}
[c]{c}%
1\\
1\\
1
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
1\\
1\\
0
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
1\\
0\\
0
\end{array}
\right)  \right\}  .$ Let $T:\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$ be the
linear transformation
\[
T\left(
\begin{array}
[c]{c}%
x\\
y
\end{array}
\right)  =\left(
\begin{array}
[c]{c}%
x+y\\
x-y\\
2x
\end{array}
\right)  .
\]


\qquad a. Find $\left[  T\right]  _{\beta}^{\gamma}$.

\qquad b. Describe $N\left(  T\right)  $ and $R\left(  T\right)  $.

\qquad c. If $E_{2}$ and $E_{3}$ denote the standard bases of $\mathbb{R}^{2}$
and $\mathbb{R}^{3},$ find the change of coordinates matrices $P$ taking
$E_{2}$ to $\beta$ and $Q$ taking $\gamma$ to $E_{3}$. Compute $Q\left[
T\right]  _{\beta}^{\gamma}P$ and compare to $\left[  T\right]  _{E_{2}%
}^{E_{3}}.$

\bigskip

\bigskip

3. (Comprehensive/graduate option only) The space $F^{n}$ acts on $F^{n}$ by
left multiplication (as in matrix multiplication), in the following way: for
each $v\in F^{n},$ define $f_{v}:F^{n}\rightarrow F$ by%
\[
f_{v}\left(  x\right)  =v^{T}x,
\]
where $v^{T}$ denotes the transpose of $v$ (so $v^{T}$ is a row vector). Show
that the map $F^{n}\rightarrow\left(  F^{n}\right)  ^{\ast}$ given by
$v\rightarrow f_{v}$ is an isomorphism.


\end{document}