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


\begin{center}
  \textbf{Math 24}\\ Homework 7
\end{center}



\begin{itemize}

\item [\#6.3.2c] Find the characteristic polynomial and all
  eigenvalues and eigenvectors for $A =
  \begin{bmatrix*}[r]2&1&1\\-4&-3&0\\-2&-2&1\end{bmatrix*}$.


\item [\#6.3.4] [(a) Prove that the matrices
  $\begin{bmatrix*}[r]1&-5387621.4\\0&2\end{bmatrix*}$ and
  $\begin{bmatrix*}[r]5&3\\-4&-2\end{bmatrix*}$ are similar.

  [(b)] Prove that the matrices
  $\begin{bmatrix*}[r]1.000000001&0\\0&2\end{bmatrix*}$ and
  $\begin{bmatrix*}[r]1&0\\0&2\end{bmatrix*}$ are not similar.

  
\item [\#4.1.1c,d] Use Theorem 4.3 to show the following sets are
  linearly independent:\newline
  
  (c) $\left\{\begin{bmatrix*}[r]2&0&1\\0&-3&4\end{bmatrix*},
    \begin{bmatrix*}[r]0&2&6\\-1&2&0\end{bmatrix*},
    \begin{bmatrix*}[r]-2&3&0\\6&0&1\end{bmatrix*} \right\}\subset
  M_{2\times 3}(\R)$.  \newline

  (d) $\{5x^2-6x+1, x^2-2x+1,10x^2-8x+1\}\subset C([0,1])$.

\item [\#4.1.12] Let $V$ be an inner product space.

  \begin{itemize}
  \item [(a)]Show that $v=0$ if and only if $\la v,w\ra =0$ for every
    $w\in V$.
  \item [(b)] Show that $v=w$ if and only if $\la v,u\ra = \la w,u\ra$
    for every $u\in V$.
  \item [(c)] Let $S, T \in \L(V)$.  Show that $S=T$ if and only if
    $\la S(v_1),v_2\ra = \la T(v_1), v_2\ra$ for every
    $v_1, v_2 \in V$.
  \end{itemize}

\item [\#4.2.4ab] Find the coordinate representation of each of the
  following vectors with respect to the orthonormal basis given in the
  corresponding part of Exercise 4.2.2.\newline
    
  (a) $\begin{bmatrix*}[r]1\\2\\-3\end{bmatrix*}$\hspace{.5in} (b)
  $\begin{bmatrix*}[r]5&7\\7&2\end{bmatrix*}$

\item[\#4.2.6ab] Find the matrix representing each of the following
  linear maps with respect to the orthonormal basis given in the
  corresponding part of Exercise 4.2.2.\newline

  (a) $T \begin{bmatrix*}x\\y\\z\end{bmatrix*}=
  \begin{bmatrix*} y\\z\\x\end{bmatrix*}$\hspace{.5in} (b)
  $T(A) = A^T$.


\end{itemize}

\end{document}


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