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


\begin{center}
  \textbf{Math 24}\\ Homework 2
\end{center}



\begin{itemize}


\item [\#2.1.5] Let $C(\R)$ be the vector space over $\R$ of
  continuous functions $f:\R\to \R$.  Define $T:C(\R) \to C(\R)$ by
  $[Tf](x) = f(x) \cos x$.  Show that $T$ is a linear map.

\item [\#2.1.7] Verify that $\colvec{4}{1}{-1}{1}{-1}$ is an
  eigenvector of $
  \begin{bmatrix}
    1&2&3&4\\2&3&4&1\\3&4&1&2\\4&1&2&3
  \end{bmatrix}$, and determine the corresponding eigenvalue.


\item [\#2.1.9c] Find all the eigenvalues and eigenvectors (over $\C$)
  of the matrix $\begin{bmatrix} 2&-7\\0&2
  \end{bmatrix}$.




\item [\#2.1.11] Find all eigenvalues and eigenvectors of the
  orthogonal projection $P\in \L(\R^2)$ described in Exercise 2.1.2.

  
\item [\#2.1.12] ($n=2,3$ only) Find all the eigenvalues and
  eigenvectors of the matrix $A\in M_n(\R)$ with $a_{ij}=1$ for all
  $i,j$, \textbf{and} $n=2,3$.

\item [\#2.1.13] Consider a linear system in matrix form: $A\x = \b$
  where $A \in M_{m\times n}(\F)$ and $\b \in \F^m$ are given.
  Suppose that $\x,\y \in \F^m$ are both solutions.  Under what
  condition is $\x+\y$ also a solution?
    
\item [\#2.1.14] Suppose that $A \in M_n(\F)$ and that for each
  $j= 1, \dots, n$, $\e_j$ is an eigenvector of $A$.  Prove that $A$
  is a diagonal matrix.

\item [\#2.2.2] Give an explicit isomorphism between $\R^2$ and the
  set of solutions of the linear system (over $\R$)
  \begin{align*}
    w-x+3z&=0\\w-x+y+5z&=0\\2w-2x-y+4z&=0.
  \end{align*}

   
\item [\#2.2.7] Find the matrix of the linear map $T:\R^2\to\R^2$
  which first reflects across the $y$-axis, then rotates
  counterclockwise $\pi/4$ radians, then stretches by a factor of 2 in
  the $y$ direction.

\item[\#2.2.8] Show that if 0 is an eigenvalue of $T \in \L(V)$, then
  $T$ is not injective.
    
\end{itemize}

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