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


\begin{center}
  \textbf{Math 24}\\ Homework 6
\end{center}



\begin{itemize}


\item [\#3.5.12]
  \begin{itemize}
  \item [(a)] Show that $\B=\{1, x, \frac{3}{2}x^2 -\frac{1}{2}\}$ is a
    basis for $\P_2(\R)$.
    \item [(b)] Find the coordinate representation of $x^2$ with
      respect to $\B$.
      \item[(c)] Let $D: \P_2(\R) \to \P_2(\R)$ be the derivative
        operator. Find the coordinate representation of $D$ with
        respect to $\B$.
        \item [(d)] Use your answers to the last two parts to compute $\frac{d}{dx}(x^2)$.
  \end{itemize}

  \item[\#3.6.4]Consider the basis $\B=\left\{\colvec{2}{2}{-3},
      \colvec{2}{-1}{2}\right\}$ of $\R^2$.

    \begin{itemize}
    \item [(a)] Find the change of basis matrices $[I]_{\B,\E}$ and
      $[I]_{\E,\B}$.
      \item [(b)] Use these change of basis matrices to find the
        coordinate representation of each of the following vectors
        with respect to $\B$.
        \[(i)\colvec{2}{1}{0}\quad (ii) \colvec{2}{0}{1}\quad (iii)
          \colvec{2}{2}{3}\quad
          (iv) \colvec{2}{4}{-5}\quad (v) \colvec{2}{-6}{1}
          \]
    \end{itemize}

    \item [\#8a,b(iii)] Consider the basis $\B=\left\{
        \begin{bmatrix}2\\3\end{bmatrix}, \begin{bmatrix}3\\5\end{bmatrix} 
\right\}$ of $\R^2$, and $\sC =
\left\{\begin{bmatrix}1\\1\\0\end{bmatrix},
\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}
\right\}$ of $\R^3$, and the linear maps $S\in \L(\R^2,\R^3)$ and
$T\in \L(\R^3,\R^2)$ given (wrt standard bases) by
\[[S]_{\E,\E} =
  \begin{bmatrix*}[r]
    2&-1\\5&3\\-3&2
  \end{bmatrix*} \qquad \mbox{and}\qquad [T]_{\E,\E}=
\begin{bmatrix*}[r]
  1&-1&1\\1&1&-1
\end{bmatrix*}.
\]

\begin{itemize}
\item [(a)] Find the change of basis matrices $[I]_{\B,\E}$ and
  $[I]_{\E,\B}$ for $\R^2$, and $[I]_{\sC,\E}$ and
  $[I]_{\E,\sC}$ for $\R^3$.
\item[(8biii)] Use these change of basis matrices to find the
  coordinate representation of 
(iii) $[S]_{\B,\sC}$.
\end{itemize}

\item [\#3.6.10] Consider the bases $\B= \left\{
    \begin{bmatrix*}[r]
      2\\-1\\0
    \end{bmatrix*},
    \begin{bmatrix*}[r]
      0\\3\\-2
    \end{bmatrix*}
\right\}$ and 
  $\sC= \left\{
    \begin{bmatrix*}[r]
      1\\1\\-1
    \end{bmatrix*}
    \begin{bmatrix*}[r]
      3\\0\\-1
    \end{bmatrix*}
\right\}$ of the subspace $U$ of $\R^3$ defined by $x+2y+3z=0$, and
let $T\in \L(U)$ be given by $T
\begin{bmatrix}
  x\\y\\z
\end{bmatrix}=
\begin{bmatrix}
  9z\\3y\\x
\end{bmatrix}$.



\begin{itemize}
\item [(a)] Find the matrix $[T]_\B$.
\item [(b)]  Find the change of basis matrices $[I]_{\B,\sC}$ and
  $[I]_{\sC,\B}$.
\item[(c)] Use your answers to the previous parts to find $[T]_{\sC}$.
\end{itemize}


\item [\#3.6.14] Consider the bases $\B=\{v_1, \dots, v_n\}$ and
  $\sC=\{w_1, \dots, w_n\}$ of $\F^n$ from the example on page 201.

  \begin{itemize}
  \item [(a)] Find the change of basis matrix $[I]_{\sC,\B}$.


  \end{itemize}

\item[\#3.6.16] Prove that the matrices 
\[
  \begin{bmatrix*}[r]
    2&0&-1\\-1&3&2\\1&3&1
  \end{bmatrix*}\mbox{ and }
\begin{bmatrix*}[r]
    2&0&-1\\2&-3&-1\\1&3&1
  \end{bmatrix*}
\]
are not similar to each other.

\item[\#3.6.18] Show that the matrix $
  \begin{bmatrix*}[r]
    5&6\\-1&-2
  \end{bmatrix*}$ is diagonalizable.


\item[\#6.1.8] Let $R: \R^3 \to \R^3$ denote reflection across the
  plane $x+2y+3z=0$.  What is $\det R$?

\item [\#6.1.9] Suppose that $\dim V = n$ and $T\in \L(V)$ has $n$
  distinct eigenvalues $\lambda_1, \dots, \lambda_n$.  Prove that
  $\det T = \lambda_1\cdots \lambda_n$.

\item [\#6.2.2c] Find the determinant of
  $\begin{bmatrix*}[r]
    1&-2&3&0\\-2&1&0&-1\\3&-4&5&-2\\0&-1&2&-3
  \end{bmatrix*}$.

\end{itemize}

\end{document}


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