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%\prof{George McNinch}
\prof{Peter Cholak}
\class{Math 221}
\quizdate{Nov 18,1996}
\examname{Extra Credit Homework}

\begin{document}
\heading

Due Friday November 22. Each problem is worth {\it one} point of
extra credit.  Consider the inner product space $V =
\mathbb{R}^4$ with the usual Euclidean inner product.  Let $W$ be
the subspace spanned by the rows of the matrix $A$.

\begin{equation*}
  A=
  \begin{bmatrix}
    \frac{1}{3} & \frac{2}{3} & 0 & \frac{2}{3}  \\ \\
    0 & 3 &  -1 & \frac{3}{2}  \\
  \end{bmatrix}
\end{equation*}

\begin{problem}
  Find a basis for $W$. What is the dimension of $W$?
\end{problem}

\begin{problem}
  \label{w-basis}
  Using the {\em Gram-Schmidt} process, find an {\em orthonormal
    basis} for the subspace $W$.
\end{problem}

\begin{problem}\label{1}
Let $v = (1,0,0,0)$.  Compute the {\em orthogonal projection}
$\text{Proj}_W(v)$ of $v$ on the subspace $W$.  

We know from a theorem in class that {\em any} vector $v$ in
$\mathbb{R}^5$ has a decomposition
\begin{equation*}
  v = w_1 + w_2
\end{equation*}
where $w_1$ is in $W$ and $w_2$ is in $W^\perp$.  Let $w_1 =
\text{Proj}_W(v)$ and let $w_2 = v - w_1$.  Verify that $w_2$ is
in $W^\perp$.  Observe (!!) that $v = w_1 + w_2$.
\end{problem}

\begin{problem}\label{2}
  Repeat the previous part for the vector $v' = (0,1,0,0)$. (i.e.
  find $w_1'$ in $W$ and $w_2'$ in $W^\perp$ so that $v' = w_1' +
  w_2'$.)
\end{problem}

\begin{problem}
  Since $V$ has dimension 4, and $W$ has dimension 2, it follows
  that $W^\perp$ has dimension 2.  Why?
\end{problem}

\begin{problem}
  Let $w_2$ and $w_2'$ be the vectors in $W_\perp$ that you found
  in problems ~\ref{1} and ~\ref{2}.  Check that these vectors
  form a basis for $W^\perp$.
\end{problem}

\begin{problem}
  Regard the vectors $w_2$ and $w_2'$ as column vectors and
  compute the products $A w_2$ and $A w_2'$.  Explain your
  results.
\end{problem}

\begin{problem}
  \label{w-perp-basis}
  Starting with the basis for $W^\perp$ that you just found, find
  an {\em orthonormal basis} for the subspace $W^\perp$ (use
  Gram-Schmidt again).
\end{problem}

\begin{problem}
  Let $S$ be the set obtained by combining the bases from
  problems \ref{w-basis} and \ref{w-perp-basis}.  Then $S$ is a
  basis of $V$.  Why?
\end{problem}

\begin{problem}
  Let $x = (a,a,b,b)$ where $a$ and $b$ are real numbers.  Find
  the co-ordinate vector $x_{(S)}$.  Compute the orthogonal
  projections $\text{Proj}_W(x)$ and $\text{Proj}_{W^\perp}(x)$.
  Verify that $x = \text{Proj}_W(x) +\text{Proj}_{W^\perp}(x).$
\end{problem}

\end{document}

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