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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Retest Exam 2 -- March 31, 2004}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 regular points
and 10 Extra Credit points.  Notation:
\item{$\bullet$}  $P_n({\bf R})$ is the vector space of 
polynomials of degree $\le n$.
\item{$\bullet$} $P({\bf R})$ is the vector space
of all polynomials (no restriction on degree).
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I.  
\item{A)}  (15) Let $V$ be a vector space with a finite
spanning set $S$.  If $T$ is a linearly independent subset
of $V$, show that $|T| \le |S|$.
\item{B)}  (10)  Use part A to deduce that if $V$ be a vector 
space with a finite spanning set, then all bases of $V$
have the same number of elements.
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II.  Let $T : {\bf R}^4 \to P_2({\bf R})$ be defined 
by 
$$T(a_1,a_2,a_3,a_4) = (a_1 + a_2 - a_4) + (a_2 - 3a_3)x + 
(-a_1 - 2a_2 + 3a_3 + a_4)x^2.$$
\item{A)} (10)  Show that $T$ is linear.
\item{B)} (15) Find the matrix of $T$ with respect to $\alpha =$
the standard basis in ${\bf R}^4$ and $\beta = \{1+x,x,x^2-1\}$
in $P_2({\bf R})$.
\item{C)} (10) Find a basis for, and the dimension of, ${\rm Ker}(T)$.
\item{D)} (10)  What is the dimension of ${\rm Im}(T)$?  Is 
$T$ surjective?
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III.  (True-False)  For each true statement, give a short proof.
For each false statement, give a reason or a counterexample.  
\item{A)} (10)  If $T : V \to W$ is linear, $\dim(V) = 5$, 
$\dim(W) = 3$, and $S = \{{\bf v}_1,{\bf v}_2,{\bf v}_3\} \subset {\rm Ker}(T)$, then
$S$ is linearly dependent.
\item{B)} (10)  Let $\beta$ be the standard basis in ${\bf R}^2$.
If $S = R_{2\pi/3}$ and $T=R_{4\pi/3}$ are the counter-clockwise
rotations about the origin, then 
$$[ST]_\beta^\beta = \pmatrix{1 & 0\cr 0& -1\cr}.$$
\item{C)} (10)  If $T : V \to W$ is linear and $\dim {\rm Ker}(T) = 0$, 
then $T$ is injective.
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{\it Extra Credit} (10)  Let $T$ be the mapping from question I, and 
let $S: P_2({\bf R}) \to P_2({\bf R})$ be the mapping defined 
by $S(p(x)) = xp'(x)$.   What is the matrix of $ST$
with respect to $\alpha = $ standard basis in ${\bf R}^4$ and
$\beta$ as in question II part B in $P_2({\bf R})$?  
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