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\noindent{\large \textbf{MATH 413/513 \hfill}}

\vspace{2 cm}

\centerline{\large{$\mathcal{EXAM}$ 1}}

\bigskip

\centerline{\large{\it October $14^{\mbox{th}}$, 2015}}

\bigskip\bigskip\bigskip

\noindent\textbf{{\large Your Name:}} \underline{\hspace{5 in}}

\bigskip\bigskip

\noindent\textbf{Directions:}

\bigskip

a. You may NOT use your book or your notes.

b. Please ask for extra scrap paper if needed.

c. Show \underline{all} work. Unless otherwise noted, a solution without work
is worth nothing.

d. Good Luck!

\vspace{.8 in}

\noindent\textbf{{\large Score:}}

\bigskip

\begin{center}%
\begin{tabular}
[c]{ll}%
1. & \underline{\hspace{1in}}\\
& \\
2. & \underline{\hspace{1in}}\\
& \\
3. & \underline{\hspace{1in}}\\
& \\
4. & \underline{\hspace{1in}}\\
& \\
5. & \underline{\hspace{1in}}\\
& \\
Total & \underline{\hspace{1in}/100}%
\end{tabular}

\end{center}

\newpage\noindent\textbf{1. (15pts) } Let $V$ be a vector space (not
necessarily finite-dimensional) and let $T:V\rightarrow V$ be a linear
transformation. For each of the following, determine whether or not the
condition implies that $T$ is (i) one-to-one, (ii) onto, and/or (iii) an
isomorphism (note:\ an isomorphism from $V$ to itself is called an
automorphism). Give short justifications.

\bigskip



\noindent\textbf{a.} $R\left(  T\right)  =V.\vspace{2.5in}$

\noindent\textbf{b.} $\operatorname{nullity}\left(  T\right)  =0$ and $V$ is
finite dimensional.$\vspace{2.5in}$

\noindent\textbf{c.} $V$ is finite dimensional and $\operatorname{nullity}%
\left(  T\right)  +\operatorname{rank}\left(  T\right)  =\dim V.$

\bigskip\newpage

\noindent\textbf{2. (20pts) }Let $V$ be a vector space, let $T:V\rightarrow V$
be a linear transformation, and let $T_{0}:V\rightarrow V$ denote the zero
transformation (that is, $T_{0}\left(  v\right)  =\vec{0}$ for all $v\in V$).
Prove that $T^{2}=T_{0}$ if and only if $R\left(  T\right)  \subseteq N\left(
T\right)  .$

\bigskip

\bigskip\newpage

\noindent\textbf{3. (20pts)} Consider the linear transformation $L_{A}%
:\mathbb{R}^{4}\rightarrow\mathbb{R}^{5}$ determined by the matrix
\[
A=\left(
\begin{array}
[c]{cccc}%
1 & 1 & 1 & 1\\
0 & 1 & 1 & 1\\
0 & 0 & 1 & 1\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{array}
\right)
\]
Let $E_{n}$ denote the standard basis for $\mathbb{R}^{n}$ and $\beta=\left\{
\left(
\begin{array}
[c]{c}%
1\\
0\\
0\\
0
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
1\\
1\\
0\\
0
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
1\\
1\\
1\\
0
\end{array}
\right)  ,\left(
\begin{array}
[c]{c}%
1\\
1\\
1\\
1
\end{array}
\right)  \right\}  $ be an ordered basis of $\mathbb{R}^{4}.$

\bigskip

\noindent\textbf{a. (10pts)} Compute $\left[  L_{A}\right]  _{\beta}^{E_{5}}.$ 

\vspace{3in}

\noindent\textbf{b. (10pts)} Compute the change of basis matrix $Q$ taking
$\beta$ to $E_{4}$ and double check that $\left[  L_{A}\right]  _{\beta
}^{E_{6}}=AQ.$

\bigskip

\bigskip\newpage

\noindent\textbf{4. (25pts)} 

\noindent\textbf{a. (15pts)} Let $V_{1}$ and $V_{2}$ be subspaces of $V,$ a
finite dimensional vector space. Show that $\dim\left(  V_{1}+V_{2}\right)
\leq\dim V_{1}+\dim V_{2}$\vspace{3.5in}

\noindent\textbf{b. (10pts)} Let $T,U:V\rightarrow W$ be linear
transformations. Prove that $R\left(  T+U\right)  \subseteq R\left(  T\right)
+R\left(  U\right)  $

\bigskip

\bigskip\newpage

\noindent\textbf{5. (20pts)} An $n\times n$ matrix is\emph{ unit upper
triangular} if it has ones along the diagonal and zeroes below the diagonal,
so it has the form%
\[
\left(
\begin{array}
[c]{ccccc}%
1 & a_{12} & a_{13} & \cdots & a_{1n}\\
0 & 1 & a_{23} & \cdots & a_{2n}\\
0 & 0 & \ddots & \ddots & \vdots\\
\vdots & \vdots & \ddots & 1 & a_{\left(  n-1\right)  n}\\
0 & 0 & \cdots & 0 & 1
\end{array}
\right)
\]


\noindent\textbf{a. (5pts)} Explain why the set of all unit upper triangular
matrices does not form a subspace of the vector space of $n\times n$ matrices.

\vspace{2.5in}

\noindent\textbf{b. (15pts)} Prove that any unit upper triangular matrix is invertible.


\end{document}