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\begin{document}

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\lhead{\bf HOMEWORK 9}
\chead{\bf  DUE:  Wed., May 30}

\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
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3& 2 & \\ \hline
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4& 0.5 & \\ \hline
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5& 0.5 & \\ \hline
& & \\
6& 1 & \\ \hline
& & \\
7& 1 & \\ \hline
& & \\
8& 1 & \\ \hline
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9& 1 & \\ \hline
& & \\
10& 1 & \\ \hline
& & \\
Extra Credit & 1 & \\ \hline
\hline
& & \\
Total& 10 & \\
\hline
\end{tabular}}

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{\bf Problem 1:} (1 point)  Let $V$ and $W$ be vector spaces over a field $F$ and $T:V \to W$ a linear transformation.  

{\bf (a)} (0.5 points) Show that $T\left({\bf 0}_V\right) = {\bf 0}_W$, where ${\bf 0}_V$ and ${\bf 0}_W$ are the additive identities of $V$ and $W$ respectively.  {\it Hint:  Use the definition of the additive identity}.

{\bf (b)} (0.5 points) Show that $T(-{\bf v}) = -T({\bf v})$, where $-{\bf v}$ is the additive inverse of ${\bf v}$ in $V$ and $-T({\bf v})$ is the additive inverse of $T({\bf v})$ in $W$.  {\it Note, you CANNOT simply say that $T(-{\bf v}) = -T({\bf v})$ by the definition of linear transformation because we do not know that `-1' is an element of our field $F$.}

{\bf Problem 2:} (1 point) Let $V$ and $W$ be vector spaces over a field $F$ and $T:V \to W$ a linear transformation.  Show that $T(V)$ is a subspace of $W$.

{\bf Problem 3:} (2 points) Let $V$ and $W$ be linearly isomorphic, finite dimensional vector spaces over $F$.  Prove 

{\bf (a)} (0.5 point) $T^{-1}:W \to V$ is a linear transformation.

{\bf (b)} (0.5 point) dim$V$ = dim$W$.

{\bf (c)} (1 point) if $\{ {\bf v}_1, {\bf v}_2, ..., {\bf v}_n\}$ is a basis for $V$, then $\left\{ T({\bf v}_1), T({\bf v}_2),...,T({\bf v}_n) \right\}$ is a basis for W.

{\bf Problem 4:}  (0.5 points) Let $V$ be a vector space over the field $F$.  If $R,S,T \in \mathcal{L}(V)$, Prove that $R \circ (S+T) = (R\circ S)+(R \circ T)$

{\bf Problem 5:}  (0.5 points) Let $V$ be a vector space over the field $F$.  Let $I \in \mathcal{L}(V)$ be defined by $I({\bf v}) = {\bf v} $ for all ${\bf v} \in V$.  Show that $T \circ I = I \circ T  = T$ for all $T \in \mathcal{L}(V)$.  

{\bf Problem 6:} (1 point)  Let $\mathbb{R}[x]$ be the vector space of polynomial functions in one variable over $\mathbb{R}$.  Define the multiplication of polynomials in the usual way.  Show that $\mathbb{R}[x]$ is a commutative algebra with identity.

{\bf Problem 7:} (1 point) Show that the number of elements in $S_n$ is $n!$.

{\bf Problem 8:} (1 point) Show that the composition of two elements of $S_n$ is also an element of $S_n$.   

{\bf Problem 9:} (1 point) Calculate the sign of all elements of $S_3$ using definition 2.4.4.

{\bf Problem 10:} (1 point) Show that any $\sigma \in S_n$ can be decomposed into the composition of transpositions.

{\bf Extra Credit 1:} (1 point) Show that $\sigma \in S_n$ is an odd permutation if and only if it is the composition of an odd number of transpositions. {\it (You may assume without proof that a transposition is an odd permutation.)} 


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