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\begin{flushright}
  Your Name Goes Here
\end{flushright}


\begin{center}
  \textbf{Math 24}\\ Homework 5
\end{center}



\begin{itemize}


\item [\#3.3.9] Suppose that $T \in \L(V,W)$, and that $V$ is
  finite-dimensional.  Prove that\newline $\dim \range(T) \le \dim V$.

\item [\#3.3.12] Suppose that $\dim V = n$ and that $T\in \L(V)$.
  Prove that $T$ has at most $n$ distinct eigenvalues.

\item [\#3.3.13] Suppose that $A \in M_n(\F)$ has $n$ distinct
  eigenvalues.  Show there is a basis of $\F^n$ consisting of
  eigenvectors for $A$.


\item [\#3.4.6] Prove that if $T\in \L(V,W)$ and $W$ is
  finite-dimensional, then $T$ is surjective if and only if
  $\rank(T) = \dim W$.

\item [\#3.4.8] Prove that if $A \in M_{m\times n}(\F)$ has rank $r$,
  then there exists $v_1 \dots, v_r\in \F^m$ and
  $w_1, \dots, w_r\in~\F^n$ such that $A = \sum_{i=1}^r v_iw_i^T$.
  Author's hint: Write the columns of $A$ as linear combinations of
  the basis $\{v_1, \dots, v_r\}$ of $C(A)$.  My hint: Do Quick
  Exercise \#16 for insight.

\item [\#3.4.11] Suppose that $Ax=b$ is a $5\times 5$ linear system
  which is consistent, but does not have a unique solution.  Prove
  that there must be a $c \in \F^5$ so that the system $Ax=c$ is
  inconsistent.

\item [\#3.4.12] Suppose that, for a given $A\in M_3(\R)$ there is a
  plane $P$ passing through the origin in $\R^3$ such that the linear
  system $Ax=b$ is consistent if and only if $b \in P$. Prove that the
  set of solutions to the homogeneous system $Ax=0$ is a line through
  the origin in $\R^3$.

\item [\#3.5.10] Let $P$ be the plane
  \[\left\{
      \begin{bmatrix}
        x\\y\\z
      \end{bmatrix}\in \R^3\Bigg\vert 4x+y-2z=0\right\}.
  \]
  \begin{itemize}
  \item [(a)] Find a basis for $P$.
  \item [(b)] Determine whether each of the following vectors is in
    $P$, and if so give its coordinate representation in terms of your
    basis.
    \[(i)
      \begin{bmatrix}
        1\\1\\1
      \end{bmatrix}, \quad (ii)
      \begin{bmatrix}
        0\\2\\1
      \end{bmatrix},\quad (iii)
      \begin{bmatrix}
        1\\-2\\1
      \end{bmatrix}.
    \]
  \end{itemize}

\item[\#3.5.15] Show that the projection $P\in\L(\R^3)$ onto the
  $xy$-plane is diagonalizable.


\item [\#3.5.16] Let $L$ be a line through the origin in $\R^2$, and
  let $P \in \L(\R^2)$ be the orthogonal projections onto $L$.  (see
  Exercise 2.1.2.)  Show that $P$ is diagonalizable.
\end{itemize}

\end{document}


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