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\begin{document}
March 9, 1995 \hfill Math 311, Exam 1 \hfill %Name:
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\sk
\q{35}Let $\ds A=\left[\begin{array}{rrrrr}
10&30&3&0&35\\8&24&-3&0&1\\10&30&3&-6&-7\\4&12&0&0&8
\end{array}\right]$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find a basis for the null space of $A$.
\item Find a basis for the column space of $A$.
\item Describe the soluton set of the equation $A\vec{x} = \vec{b}$,
where $\vec{b} = (-2,20,52,4)$.
\item Describe the solution set of the equation $A\vec{x} = \vec{b}$,
where $\vec{b} = (0,1,0,1)$.
\end{list}
\sk
\q{10}Find an orthonormal basis of $R^3$ such that the first two
vectors in your basis form a basis for the plane $x+y+2z=0$.
\sk
\q{10}Find the matrix representation with respect to the standard
basis of the linear transformation that projects $R^3$ onto the plane
$x+y+2z=0$. This linear transformation leaves every point in the plane
fixed and maps points not on the plane orthogonally to the plane.
\sk
\q{20}Let $\ds A=\frac13\left[
\begin{array}{rrr}1&0&-2\\1&2&2\\1&0&4\end{array}\right]$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find the eigenvalues and eigenvectors of the matrix $A$.
\item Find a diagonal matrix $D$ and a nonsingular matrix $P$ for
which $A=PDP^{-1}$.
\item Compute $A^{149}$.
\item $\ds \lim_{n\to\infty}A^n =$.
\end{list}
\sk
\q{15}Define the function $L:P_2\rightarrow R^4$ by
$L(p)=(p(2),p(-1),p(1),p(2))$.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Find the matrix representation of $L$ with respect to the bases
$\{\,1,\ t,\ t^2\,\}$ and $\{\,\vec{e}_1,\ \vec{e}_2,\ \vec{e}_3,\
\vec{e}_4\,\}$ of $P_2$ and $R^4$ respectively.
\item Find all vectors in $\vec{p}\in P_2$ for which $L(\vec{p}) = (1,0,1,1)$.
\item Is the point $(2,3,-5,-1)$ in the image of $L$
\end{list}
\sk
\q{10}Define each of the following terms, and give an example of each.
No example, no credit.
\begin{list}{\alph{bean}. }{\usecounter{bean}}
\item Coordinates
\item Linear function
\end{list}
\end{document}