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\begin{document}
Oct. 12, 1995 \hfill Math 311, Exam 1 \hfill Name:
\underline{\makebox[1.75in]{}}
\Sk
\q{20}Let $A=\left[\begin{array}{rrrr}1&6&-7&8\\2&1&-14&-6\\4&3&-28&-10
\end{array}\right]$
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Show that the solution set to the system of equations
$A\vec{x}=[0,0,0]^T$ is a subspace. Be sure to say of what vector space
the solution set is a subspace.
\item Describe the solution set to the system of equations
$A\vec{x}=[8,-17,-31]^T$. 
\item Describe the solution set to the system of equations
$A\vec{x}=[1,1,1]^T$. 
\end{list}
\sk
\q{25} Let $\ds
A=\left[\begin{array}{rr}5&1\\1&6\end{array}\right]$. 
Define the following inner product on $R^2$:
$<\vec{x},\vec{y}>=A\vec{x}\cdot\vec{y}$, where the dot on the right
side of the equals sign signifies the usual scalar product in $R^2$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Verify that this is an inner product.
\item Find the length of the vector $\vec{x}=[1,0]$.
\item Find an orthonormal basis of $R^2$.
\end{list}
\sk
\q{30}Let $f(t)=a_0 + a_1\sin(t) + b_1\cos(t) +
a_2\sin(2t)+b_2\cos(2t)$. Let
\hfill\break
$D=\{\,(-2,1),\,(-1,1),\,(0,1),\,(1,0.5),\,(2,2),\,(3,1)\,\}$. It is
not possible to find
constants $a_i$ and $b_i$ so that the graph of $f(t)$ passes through
the given data points $D$. Solve the associated least squares problem
by solving the normal equations via a QR factorization of the
corresponding coefficient matrix. Be sure to write down what system of
equations you are finding a least squares solution to.
\sk
\q{20}Let $B_1=\left\{\,
\left[\begin{array}{rr}1&0\\1&1\end{array}\right],
\ \left[\begin{array}{rr}1&1\\0&1\end{array}\right],
\ \left[\begin{array}{rr}1&0\\0&1\end{array}\right],
\ \left[\begin{array}{rr}0&0\\1&1\end{array}\right]\,\right\}$, and
let 
\sk
$B_2=\left\{\,
\left[\begin{array}{rr}2&0\\0&0\end{array}\right],
\ \left[\begin{array}{rr}0&-1\\1&0\end{array}\right],
\ \left[\begin{array}{rr}1&0\\-3&0\end{array}\right],
\ \left[\begin{array}{rr}0&1\\0&-1\end{array}\right]\,\right\}$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Both $B_1$ and $B_2$ are bases of $M_{2,2}$ Show that $B_1$ is a
basis of $M_{2,2}$. Do not worry about showing that $B_2$ is a basis.
\item If $[\vec{x}]_{B_1} = [2,1,-3,0]$, what does $\vec{x}=$?
\item Find the coordinates of
$\left[\begin{array}{rr}1&0\\0&1\end{array}\right]$ with respect to
the basis $B_2$.
\item Find the change of basis matrix $P$ such that
$[\vec{x}]_{B_2}=P[\vec{x}]_{B_1}$. 
\end{list}
\sk
\q{5}Let $A$ be an $m\times n$ matrix. Suppose that $A\vec{x}=\vec{0}$
for every $\vec{x}\in R^n$. Show that $A$ must be the $m\times n$ zero matrix.


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