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\pagestyle{fancy} \lhead{{\sf Homework 1 $|$ Due August 29 (Wednesday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Fall 2012 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 1.1, 1.2, and 2.1. \\


\begin{enumerate}

  % Made up myself
\item For each vector $\v$, compute its \emph{norm},
  $||\v||=(\v\cdot\v)^{1/2}$, and then \emph{normalize} it, by
  computing $\v/||\v||$.
  \[
  \u=\begin{bmatrix} 3 \\ 4\end{bmatrix}\,,\qquad
  \v=\begin{bmatrix} 1 \\ 1 \end{bmatrix}\,,\qquad
  \w=\begin{bmatrix} 2 \\ -1 \\ 3\end{bmatrix}\,.\qquad
  \]



  % Made up myself
\item For a unit vector $\n$, the \emph{projection} of $\v$ onto $\n$ is
  the quantity $\v\cdot\n$. This measures the magnitude of $\v$ in the
  $\n$-direction. Consider the following four unit vectors:
  \[
  \e_1=\begin{bmatrix} 1 \\ 0\end{bmatrix}\,,\qquad
  \e_2=\begin{bmatrix} 0 \\ 1\end{bmatrix}\,,\qquad
  \n_1=\begin{bmatrix} \sqrt{2}/2 \\ \sqrt{2}/2\end{bmatrix}\,,\qquad
  \n_2=\begin{bmatrix} -\sqrt{2}/2 \\ \sqrt{2}/2\end{bmatrix}\,.
  \]
  \begin{enumerate}
  \item Draw the vectors $\{\e_1,\e_2\}$ in $\R^2$, and sketch the
    square ``grid'' that they determine. Do the same thing for
    $\{\n_1,\n_2\}$ but on a new set of axes.
  \item Write the vector $\w=\begin{bmatrix} 3 \\ 4\end{bmatrix}$ as a
    \emph{linear combination} of $\{\e_1,\e_2\}$. That is, write
    $\w=a_1\e_1+a_2\e_2$ and determine $a_1,a_2\in\R$. Then, write
    $\w$ as a linear combination of $\{\n_1,\n_2\}$.
  \item Sketch $\w$ on both sets of axes, and show how these sketches
    match your answers to Part~(b). 
  \item The $2\times 2$ matrix $\A=[\n_1\;\;\n_2]$ can be thought of
    as a \emph{linear map}, $\A\colon\R^2\to\R^2$. Describe this
    linear map (geometrically) in a sentence. [\emph{Hint}: You can
      think of the ``input'' as one of your grids, and the ``output''
      as the other grid.]
  \end{enumerate}

  

  % Made up myself
\item A set $\{\v_1,\dots,\v_n\}$ of vectors is \emph{orthogonal} (or
  \emph{perpendicular}) if $\v_i\cdot\v_j=0$ for all $i\neq j$. The
  set is furthermore \emph{orthonormal} if each $\v_i$ is a unit
  vector. That is, if
  \[
  \v_i\cdot\v_j=\begin{cases} 0 & \mbox{if $i\neq j$} \\
  1 & \mbox{if $i=j$.}\end{cases} \\
  \]
  \begin{enumerate}
  \item Show that the set of vectors $\{\v_1,\v_2,\v_3\}$, where 
    \[
    \v_1=\begin{bmatrix} 1 \\ 2 \\ -2\end{bmatrix}\,,\qquad
    \v_2=\begin{bmatrix} 0 \\ 1 \\ 1\end{bmatrix}\,,\qquad
    \v_3=\begin{bmatrix} -4 \\ 1 \\ -1\end{bmatrix}\,.
    \]
    is an orthogonal set, but not orthonormal. 
  \item Normalize $\v_1$, $\v_2$, and $\v_3$ to get an orthonormal
    \emph{basis} of $\R^3$, $\{\n_1,\n_2,\n_3\}$.
  \item Express the vector $\w=(1,2,3)$ in terms
    of $\n_1$, $\n_2$, and $\n_3$. That is, find $a_1$, $a_2$, and $a_3$
    such that
    \[
    \w=\begin{bmatrix} 1 \\ 2
    \\ 3\end{bmatrix}=a_1\n_1+a_2\n_2+a_3\n_3\,.
    \]
    \item Express the vector $\w$ as a linear combination of $\v_1$,
      $\v_2$, and $\v_3$. That is, write $\w=b_1\v_1+b_2\v_2+b_3\v_3$
      and find a formula for each $b_i$. [\emph{Hint}: It should be in
        entirely in terms of dot products of $\v_i$ and $\w$. Start by
        substituting $\v_i/||\v_i||$ in for $\n_i$ in your answer to
        Part~(c).]
  \end{enumerate}



  % Made up myself
\item Consider the matrix $\A=[\u\;\;\v\;\;\w]=\begin{bmatrix} 1&2&3
  \\ 4&5&6 \\ 7&8&9 \end{bmatrix}$.
  \begin{enumerate}
  \item Find a non-zero linear combination $x_1\u+x_2\v+x_3\w$ of the
    column vectors of $\A$ that gives the zero vector.
  \item Describe the set of all \emph{solutions} to
    $x_1\u+x_2\v+x_3\w=\0$. [\emph{Hint}: It is a line. Which line is
    it?]
  \item Describe the set of all \emph{linear combinations}
    $x_1\u+x_2\v+x_3\w$. We say that this is the \emph{subspace} of
    $\R^3$ that is \emph{spanned} by the set $\{\u,\v,\w\}$.
  \end{enumerate}



  % Made up myself
\item Given an $n\times m$ matrix $\A=[a_{ij}]$, the \emph{transpose}
  of $\A$ is an $m\times n$ matrix defined as $\A^T=[a_{ji}]$. For
  each of the following three matrices $\M$, compute its transpose
  $\M^T$, as well as the products $\M^T\M$ and $\M\M^T$.
  \[
  \A=\begin{bmatrix} a \\ b \\ c\end{bmatrix}\,,\qquad
  \B=\begin{bmatrix} 1 & -2 \\ -1 & 3 \\ 0 & 4\end{bmatrix}\,,\qquad
  \bold{C}=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &
  \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\,.\qquad
  \]



  % Made up myself
\item Consider the following vectors $\u$, $\v$, $\w$, and the matrix
  $\A=[\u\;\;\v\;\;\w]$:
  \[
  \u=\begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix},\quad
  \v=\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix},\quad
  \w=\begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix},\qquad
  \A=\begin{bmatrix} u_1&v_1&w_1 \\ u_2&v_2& w_2 \\ u_3&v_3& w_3 \end{bmatrix}\,.
  \]
  \begin{enumerate}
  \item Write out $\A^T\A$ in terms of the \emph{dot products} of
    $\u$, $\v$ and $\w$.
  \item Write out $\A^T\A$ and in terms of the vectors $\u$, $\v$ and
    $\w$ and their transposes, but \emph{not} their individual
    entries. [Recall that $\u\cdot\v=\u^T\v$.]
  \item Give a complete characterization of which matrices $\A$ have
    the property that $\A^T\A=\I$, where $\I$ is the \emph{identity
      matrix}. Give a geometric description of your answer in terms of
    the column vectors.
  \end{enumerate}


\end{enumerate}

\end{document}
