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


\begin{center}
  \textbf{Math 24}\\ Homework 3
\end{center}



\begin{itemize}


\item [\#2.3.4] Let $T:\R^2 \to \R^2$ be the map defined by first
  rotating counterclockwise by an angle $\theta$ and then reflecting
  across the line $y=x$.  Find the matrix of $T$.

\item [\#2.3.11] Suppose that $A \in M_{m\times n}(\F)$ is
  right-invertible, meaning that there is a matrix
  $B\in M_{n\times m}(\F)$ so that $AB = I_m$.  Show that $m\le n$.
  Author's hint: Show that given any vector $\b \in \F^m$, the
  $m\times n$ linear system $A\x = b$ is consistent, and use Theorem
  1.2. There are other ways to reach the desired conclusion.

\item [\#2.4.4d,f] Determine whether the matrices are singular or
  invertible; if invertible, find the inverse.
  \[
    (d) \begin{bmatrix} 1&2&3\\4&5&6\\7&8&9
    \end{bmatrix}
    \qquad\mbox{ and }\qquad (f) \begin{bmatrix*}[r]
      1&0&2&-1\\0&-1&-2&1\\2&0&1&0\\0&2&-3&2
    \end{bmatrix*}
  \]


\item [\#2.4.6a] Compute $
  \begin{bmatrix*}[r]
    1&0&1\\3&2&0\\0&1&-2
  \end{bmatrix*}^{-1}$, and use it to solve the system
  \begin{align*}
    x+z&=1\\
    3x+2y&=2\\
    y-2z&=3
  \end{align*}.

\item [\#2.5.4d] Determine whether the list of vectors
  $\left\{\colvec{3}{1}{1}{0}, \colvec{3}{1}{0}{1},
    \colvec{3}{0}{1}{1}\right\}$ span $\R^3$.

\item [\#2.5.6d] Determine whether $\lambda =3$ is an eigenvalue for
  $A =
  \begin{bmatrix*}[r]
    7&-2&4\\0&1&0\\-6&2&-3
  \end{bmatrix*}$, and if so express $\Eig_\lambda(A)$ as the span of
  some list of vectors.

\item[\#2.5.9] It is a fact (which you do not have to prove) the every
  solution to the homogeneous differential equation
  \[\frac{d^2f}{dt^2} + f(t) = 0
  \]
  is a linear combination of $f_1(t) = \sin t$ and $f_2(t) = \cos t$.
  \begin{enumerate}[label=(\alph*)]

  \item Show that $f_p(t) = 2e^{-t}$ is a solution to the differential
    equation
    \[\frac{d^2f}{dt^2} + f(t) = 4e^{-t}.
    \]

  \item Use Proposition 2.42 to show that every solution of the
    differential equation in part (a) is of the form
    \[f(t) = 2e^{-t}+k_1\sin t + k_2 \cos t
    \]
    for constants $k_1, k_2 \in \R$.
  \item Determine all solutions $f(t)$ to the differential equation in
    part (a) which satisfy $f(0)=1$.
  \item Determine all solutions $f(t)$ to the differential equation in
    part (a) which satisfy $f(0)=a$ and $f(\pi/2) = b$.
         
  \end{enumerate}

\item[\#2.5.10] Let $T:V \to W$ be a linear map.  Prove that if $U$ is
  a subspace of $V$, then $T(U)$ is a subspace of $W$.

\item[\#2.5.15] Suppose that $\lambda \in \F$ is an eigenvalue for
  $T\in \L(V)$, and $k\ge 1$ an integer. Show that $\lambda^k$ is an
  eigenvalue for $T^k$, and
  $\Eig_\lambda(T) \subseteq \Eig_{\lambda^k}(T^k)$.

\item[\#2.5.16] Give an example of a linear map $T: V\to V$ with
  eigenvalue $\lambda$ so that
  $\Eig_\lambda(T) \ne \Eig_{\lambda^2}(T^2)$.
\end{itemize}

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