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\begin{document}

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\lhead{\bf HOMEWORK 8}
\chead{\bf  DUE:  Mon., May 21}

\rhead{\bf  NAME: \hspace{80} }
\noindent

\cfoot{\thepage}
 {\bf DIRECTIONS:}  

\begin{itemize}                 
         \item Turn in your homework as {\bf SINGLE-SIDED} typed or handwritten pages.  
         \item {\bf STAPLE} your homework together.  Do not use paper clips, folds, etc.
	\item {\bf STAPLE} this page to the front of your homework.
         \item {Be sure to write your name on your homework.}
	\item Show all work, {\bf clearly and in order}.
\end{itemize}

{\bf You will lose point 0.5 points for each instruction not followed.}

\bigskip\bigskip
\centerline{\begin{tabular}{|c||c|c|}\hline
& & \\
Questions & \quad Points \quad & \qquad Score \qquad\,\\ \hline\hline
& & \\
1 & 1 & \\ \hline
& & \\
2& 1 & \\ \hline
& & \\
3& 1 & \\ \hline
& & \\
4& 1 & \\ \hline
& & \\
5& 2 & \\ \hline
& & \\
6& 1 & \\ \hline
& & \\
7& 1 & \\ \hline
& & \\
8& 1 & \\ \hline
& & \\
9& 1 & \\ \hline
\hline
& & \\
Total& 10 & \\
\hline
\end{tabular}}

\newpage

{\bf Problem 1:}  (1 point)  Let $F$ be a field and $F^n$ be a vector space.  Prove that the set of canonical basis vectors, $S = \left\{ {\bf e}_1, {\bf e}_2,...,{\bf e}_n \right\}$, is a linearly independent set.

{\bf Problem 2:}  (1 point) Consider the vector space $V=F^n$.  Let ${\bf v} \in V\setminus \left\{ {\bf e}_1, {\bf e}_2,...,{\bf e}_n \right\}$.  Using only the definition of the canonical vectors, ${\bf e}_j$, and the definitions of linearly dependent and independent, prove that $\left\{ {\bf e}_1, {\bf e}_2,...,{\bf e}_n, {\bf v} \right\}$ is a linearly dependent set.

{\bf Problem 3:}  (1 point) Consider the set of vectors $\left\{ {\bf v}_1, {\bf v}_2,...,{\bf v_m} \right\}$.  Suppose ${\bf v}_j = {\bf 0}$ for some $j$ such that $1 \le j \le m$.  Prove that this is a linearly dependent set.

{\bf Problem 4:}  (1 point)  Let ${\bf v}_1$ and ${\bf v}_2$ be vectors in a vector space $V$.  Show that the set $\left\{ {\bf v}_1, {\bf v}_2 \right\}$ is a linearly dependent set if and only if one of these vectors is a scalar multiple of the other.

{\bf Problem 5:} (2 points)  Determine by inspection if the given set is linearly dependent.  You do not need to prove your answer.

{\bf (a)} (0.5 points) $(1,0)$, $(0,1)$, $(\sqrt{2}, \pi )$.

{\bf (b)} (0.5 points) $(1,7,6)$, $(2,0,9)$, $(3,1,5)$, $(4,1,8)$.

{\bf (c)} (0.5 points) $(2,3,5)$, $(0,0,0)$, $(1,1,8)$.

{\bf (d)} (0.5 points) $(-2, 4, 6, 10)$, $(3, -6, -9, -15)$. 

{\bf Problem 6:}  (1 point)  Let $V$ be a vector space over a field $F$.  Show $\{ {\bf 0} \}$ and $V$ are subspaces of $V$.

{\bf Problem 7:}  (1 point) Let $V=\mathbb{Q}[x]$, and let $W$ be the collection of all polynomials in $\mathbb{Q}[x]$ whose degree is less than or equal to a fixed non-negative integer $n$.  

{\bf (a)} (0.5 points)  Prove that $W$ is a subspace of $V$.

{\bf (b)} (0.5 points) Find the dimension of $W$ and justify your answer.

{\bf Problem 8:}  (1 point) Let $F$ be a field and consider the vector space $V=F^n$ and for a fixed $m \le n$, let $W= \left\{ {\bf v} \in V  | \ \textrm{v is a linear combination of the basis vectors}\ {\bf e}_1, {\bf e}_1,...,{\bf e}_m \right\}$.  

{\bf (a)} (0.5 points) Prove that $W$ is a subspace of $V$.

{\bf (b)} (0.5 points) Find the dimension of $W$ and justify your answer.

{\bf Problem 9:} (1 point) Let $V=\mathbb{R}$ be a vector field over $F=\mathbb{R}$.  Let $a \in \mathbb{R}$.  Consider $T_a : V \to V$ where $T_a(x) = ax$ for all $x \in \mathbb{R}$.  Prove that $T_a$ is a linear transformation.

\end{document}