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\begin{document}
\title{Math 241 Homework}
\author{Mary Radcliffe}
\date{Due 18 October 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $V$ be a vector space of dimension $n$. Prove that if $S=\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_{n+1}\}$, then $S$ is linearly dependent.

\item Let $A$ and $B$ be $n\times n$ matrices. Prove that $\rank{AB}\leq \rank{B}$. Give an example to show that equality can hold. Give an example to show that equality may not hold.

\item Mark each of the following as true or false. Give a reason for your answer.
\begin{enumerate}
\item If $A$ is a $3\times 5$ matrix, then the columns of $A$ must be linearly dependent.
\item If $V$ is a vector space, and $S=\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ spans $V$, then $\dim{V}\leq n$.
\item If $V$ is a vector space, and $B=\{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ is a linearly independent subset of $V$, then $\dim{V}\leq n$.
\item If $A$ has more columns than rows, then $\Null{A}$ is nontrivial.
\item If $A$ has more rows than columns, then $\Null{A}$ is nontrivial.
\end{enumerate}

\item Suppose that $V$ and $W$ are both vector spaces of dimension $n$ over the field $F$. Show that there exists an invertible linear function $f:V\to W$.

\item Use the Rank-Nullity Theorem to prove that every invertible matrix is square.

\item Spend a few minutes reading back over the Rank-Nullity Theorem, and thinking about why we might care about a theorem like this. Write 5 reasons why the Rank-Nullity Theorem could be useful in terms of deriving practical information about a matrix.
\end{enumerate}
\end{document}