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% DOCUMENT
\begin{document}
  % INTRO
    \noindent
    Your name:
    \vskip 12pt
    \noindent
    Instructor (please circle):
    \hspace{1in}
    \noindent
    Alex Barnett
    \hskip .7in
    Michael Musty
    \vskip 12pt
    \noindent
        {\bf \large Math 22 Summer 2017, Homework 8, due Fri August 18}
        \quad
    {\em Please show your work, and check your answers. No credit is given for solutions without work or justification.}
    \vskip .25in
  % NUMBERED QUESTIONS
  \thispagestyle{empty}
  \begin{enumerate}
    \item
      % Question 1
      Let
      $
      A =
      \left[
        \begin{array}{rrrr}
          5&1&-1&5\\
          1&5&5&-1\\
          -5&1&1&5
        \end{array}
      \right]
      $
      and note that the rows of $A$ are orthogonal.
      \begin{enumerate}
        \item
          Without using row reduction,
          write
          $
          \mbf{y}=
          \left[
            \begin{array}{r}
              11\\5\\3\\-1
            \end{array}
          \right]
          $
          as a linear combination of the rows of $A$.
          \vspace{3in}
        \item
          Let $W = \mathrm{Col\,}(A^T)$.
          To which fundamental subspace of the matrix $A$ is $W^\perp$ equal?
          \vspace{1in}
        \item
          What is dim $W^\perp$? Prove your answer.
      \end{enumerate}
    \pagebreak
    \item
      % Question 2
      Let
      $
      \mbf{v}_1 =
      \left[
        \begin{array}{r}
          1\\1\\-2
        \end{array}
      \right],\quad
      \mbf{v}_2 =
      \left[
        \begin{array}{r}
          1\\1\\1
        \end{array}
      \right],\quad
      \mbf{y} =
      \left[
        \begin{array}{r}
          -1\\1\\3
        \end{array}
      \right].
      $
      Let $W = \mathrm{Span}\{\mbf{v}_1,\mbf{v}_2\}$.
      \begin{enumerate}
        \item
          Find the element of $W$ whose distance to $\mbf{y}$
          is as small as possible.
          \vspace{3in}
        \item
          Compute the distance from the previous part.
          \vspace{1.5in}
        \item
          Let $U$ be the $3\times 2$ matrix whose columns are
          $\mbf{v}_1/\|\mbf{v}_1\|$ and
          $\mbf{v}_2/\|\mbf{v}_2\|$.
          {\em Without} computing any matrix-vector multiplication,
          find $U U^T \mbf{y}$ and explain why.
          \vspace{1.5in}
        \item
          Without computing a matrix-vector multiplication, compute
          the 2-norm of the vector $U \left[\begin{array}{r}4\\3\end{array}
      \right]$, with $U$ as in part (c). Explain.

      \end{enumerate}
    \pagebreak
    \item
      % Question 3
      Let
      $
      A =
      \left[
        \begin{array}{rrr}
          -1&0&0\\
          1&1&0\\
          0&1&-2\\
          0&0&1
        \end{array}
      \right].
      $
      \bigskip
      \begin{enumerate}
        \item
          Find an orthogonal basis for Col $A$ using the Gram-Schmidt algorithm.
          \vspace{3.5in}
        \item
          Compute the $QR$ factorization of $A$.
          (i.e.
          Find matrices $Q$ and $R$ so that $A = QR$,
          the columns of $Q$ form an orthonormal basis for Col $A$,
          and $R$ is an upper triangular invertible matrix
          with positive entries along its diagonal.)
\vfill
        \item [BONUS] what happens during Gram-Schmidt if the columns of $A$
          were linearly dependent?
      \end{enumerate}
  \end{enumerate}
\end{document}
