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


\begin{center}
  \textbf{Math 24}\\ Homework 8
\end{center}



\begin{itemize}

\item [\#4.1.4] Let $A, B \in M_{m\times n}(\C)$.  Show that $\la
  A,B\ra_F = \sum_{j=1}^n \la a_j, b_j\ra$ and $\|A\|^2_F = \sum_{j=1}^n \|a_j\|^2$.

\item [\#4.2.12c] Carry out Gram-Schmidt on the following set of
  vectors:
  \[
  \left\{\begin{bmatrix*}[r]1\\0\\0\\-1\end{bmatrix*},
\begin{bmatrix*}[r]0\\1\\0\\-1\end{bmatrix*},\begin{bmatrix*}[r]0\\0\\1\\-1\end{bmatrix*}
  \right\}
\]

\item [\#4.2.17] Let $A=\
  \begin{bmatrix}
    0&1&1\\1&0&1\\1&1&0
  \end{bmatrix}\in M_3(\R),$ and define a new inner product on $\R^3$
  by\newline
  $\la x,y\ra_A :=\la Ax, Ay\ra$ where the inner product on the
  right-hand side is the standard inner product on $\R^3$.  Find an
  orthonormal basis of $\R^3$ with respect to the
  $\la\cdot,\cdot\ra_A$.


\item [\#4.2.18] Let $A \in M_n(\C)$ be an invertible matrix and
  define a non-standard inner product on $\C^n$ by $\la x,y\ra_A := \la Ax,Ay\ra$
  where the inner product on the
  right-hand side is the standard inner product on $\C^n$.  For the
  standard basis vectors $\{e_1, \dots, e_n\}$ for $\C^n$,
  \begin{itemize}
  \item [(a)] Find $\la e_j, e_k\ra_A$.
    \item [(b)] Under what circumstances is the standard basis of
      $\C^n$ orthonormal with respect to $\la \cdot, \cdot\ra_A$?
  \end{itemize}

\item [\#4.3.2a] Find the matrix (wrt the standard basis) of the
  orthogonal projection onto the span of the vectors
  $\left\{[1,1,1,1]^T, [1,2,3,4]^T\right\}\subset \R^4$.

\item[\#4.3.3b] Find the matrix (wrt the standard basis) of the
  orthogonal projection onto the subspace
  \[\{[x,y,z]^T\in \R^3\mid 3x-y-5z=0\}.
  \]
  Part 8 of Theorem 4.16 is probably of use.

\item [\#4.3.5a] Find the point in the subspace $U$ which is closest
  to the point $x$, where
  \[U = \Span\{[1,0,-1,2]^T, [2,-1,1,0]^T\} \mbox{ and }x = [1,2,3,4]^T.
    \]

\item [\#4.3.5d] Find the point in the subspace $U$ which is closest
  to the point $x$, where
 \[U =
   \ker \begin{bmatrix*}[r]2&0&-1&3\\1&2&3&0\end{bmatrix*}\subseteq
   \R^4 \mbox{ and } x=[-2,1,1,2]^T.
 \]
 
 
\item [\#4.3.7] Use simple linear regression to find the line in the
  plane which comes closest to the data points $(-2,1), (-1,2),(0,5),(1,4),(2,8)$.

\item[\#4.3.14]
  \begin{itemize}
  \item [(a)] Show that $V= \{A \in M_n(\R) \mid A^T=A\}$ and $W = \{A \in
    M_n(\R) \mid A^T=-A\}$ are subspaces of $M_n(\R)$.

    \item [(b)] Show that $V^\perp = W$ where the inner product is the
      usual Frobenius inner product.

    \item [(c)] Show that for any $A \in M_n(\R)$,
      \[P_V(A) = \textrm{Re}(A) \mbox{ and } P_W(A) = \textrm{Im}(A),
      \]
      where $\textrm{Re}(A)$ and $ \textrm{Im}(A)$ are defined in
      Exercise 4.1.5 (ignore the $i$ in the denominator of $\textrm{Im}(A)$).
  \end{itemize}


  
\end{itemize}

\end{document}


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