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 \myhead{Mat 5320}{Exam 1}{Autumn '05}{}
 
 You must work alone. You may use the text, other books, your notes, etc.~and your instructor; you may not consult another person in any way. Answer all questions completely. \\[1ex]
 
 %%%%%%%%
\S I. \textsc{True} and/or \textsc{False}. There are \ref{LastTF} questions at 2 points each. Circle your answer.
\begin{enumerate}
\item \TF{The set of all complex-valued polynomials on the interval $[-1,+1]$ is a subspace of $\mathcal{C}_\mathbb{C}[-1,+1],$ the space of continuous complex-valued functions on $[-1,+1].$}

\item \TF{Let $n\in\mathbb{Z}^+\!.$ Every $n$-dimensional vector space has a basis containing the zero vector $0.$} \\

\item \TF{If $A=\{x_1, x_2, \dots ,x_n\} \subset X$ is linearly independent and $x\notin \mathcal{V}(A),$ then it follows that the set $A_x=\{x, x_1, x_2, \dots ,x_n\}$ is linearly independent.}

\item
	\TF{Let $X$ be a finite dimensional vector space. Let $\mathcal{I}$ be the collection of linearly independent subsets of $X;$ let $\mathcal{S}$ be the collection of spanning subsets of $X.$ Then $\mathcal{I}\cap\mathcal{S}\ne\emptyset.$} 
	
\item \label{LastTF} 
	\TF{The set of linear transformations $L(X,X)$ is a subspace of the space of linear functionals $X^f\!.$} \\[2ex]
\end{enumerate}

 
 %%%%%%%%%
 \S II. \textsc{Problems}. There are \ref{LastProb} problems at $20$ points each.
 
\begin{enumerate}
\item Set $Y=\{[1,0,1,0], [1,1,1,0], [3,2,1,0], [1,2,3,0]\}\subset\mathbb{R}^4.$ 
	\begin{enumerate}
	\item Demonstrate that $Y$ is linearly dependent. 
	\item Prove that $\mathcal{V}(Y)\cong \mathbb{R}^3.$
	\end{enumerate}

\item Determine which of the following are linear transformations.
	\begin{enumerate}
	\item The mapping $R:\mathbb{R}^4\rightarrow\mathbb{R}^2$ given by $R([x_1,x_2,x_3,x_4]) = [x_1+x_2, x_3+x_4].$
	\item The mapping $S:\mathbb{R}^3\rightarrow\mathbb{R}^3$ given by $S([x_1,x_2,x_3]) = [x_2,x_3,x_1].$
	\item The mapping $T:\mathbb{R}^2\rightarrow\mathbb{R}^1$ given by $T([x,y]) = x\,\tan(y).$
	\end{enumerate}

\item Suppose that the linear transformation $\phi:\mathbb{R}^3\rightarrow \mathbb{R}^3$ is given by
	$\phi([x,y,z]) = [2x, x-z, 2z].$
	\begin{enumerate}
	\item Find $\phi([1, 0, 0]),  \phi([0, 1, 0]),$ and $\phi([0, 0, 1]).$
	\item Find the range space $\mathfrak{R}(\phi)$ and the rank $\rho(\phi).$
	\item Find the null space $\mathfrak{N}(\phi)$ and the nullity $\nu(\phi).$
	\item Determine $\mathfrak{R}(\phi)\cap\mathfrak{N}(\phi).$
	\end{enumerate}

\item Prove: 
	\begin{theorem} Let $F$ be a field and $n\in\mathbb{Z}^+\!.$ Then $F^n$ is a vector space.
	\end{theorem}

\item Prove:
	\begin{theorem} Let $T\in L(X, Y).$ Then $T$ is injective if and only if $\mathfrak{N}(T) = \{0\}.$
	\end{theorem}
	
\item \label{LastProb} Prove:
	 \begin{theorem} Let $X$ be a vector space with a subspace $X_i$ for each $i\in \mathcal{I}$ for some index set $\mathcal{I}.$ Then the intersection $\mathcal{X}=\displaystyle\bigcap_{i\in\mathcal{I}} X_i$ is a subspace of $X.$ \\[2ex]
	 \end{theorem}

\end{enumerate}


\textsc{ec}: Give the author, the play, the act, and the scene for the audio clip on the web site.
 
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