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\begin{document}

\title{Chapter Check for Chapter 1}
\maketitle

1. Let $\mathcal{F}\left(  X,F\right)  $ denote the set of functions from the
set $X$ to the field $F.$

\bigskip



\qquad a. Describe the usual vector space structure for$\mathcal{F}\left(
X,F\right)  $ (you do not have to show it is a vector space).

\bigskip



\qquad b. Let $x_{0}\in X$ and for each $a\in F,$ consider the set
$S_{a}=\left\{  f\in\mathcal{F}\left(  X,F\right)  :f\left(  x_{0}\right)
=a\right\}  .$ For which values of $a$ is $S_{a}$ a subspace? Justify your answer.

\bigskip



\qquad c. If $X$ is a finite set, show that the dimension of $\mathcal{F}%
\left(  X,F\right)  $ is equal to the number of elements in $X.$

\bigskip

\bigskip



2. 

\qquad a. Find a basis for $\mathbb{R}^{3}$ that contains the vector $\left(
\begin{array}
[c]{c}%
1\\
0\\
1
\end{array}
\right)  .$ Justify your answer.

\bigskip



\qquad b. Is the set $\left\{  \left(
\begin{array}
[c]{c}%
1\\
-1\\
1
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
2\\
2\\
1
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
1\\
-5\\
2
\end{array}
\right)  \right\}  $ a basis for $\mathbb{R}^{3}$?\ Why or why not?

\bigskip



\bigskip



3. (Comprehensive/graduate option only) Consider the subspace $W=\left\{
\left(  t,t,t\right)  \in\mathbb{R}^{3}:t\in\mathbb{R}\right\}  $ of
$\mathbb{R}^{3}.$ Give an explicit basis for $\mathbb{R}^{3}/W.$ Justify your answer.
\end{document}