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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Exam 2 -- March 23, 2007}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 regular
points and 5 Extra Credit points (see III C below).
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I. (15) Let $V$ be a vector space and let $S$ be a finite
subset of $V$ with ${\rm Span}(S) = V$.  If $T$ is a linearly
independent subset of $V$, show that $|T| \le |S|$.  (Here
$|S|$ denotes the number of elements in $S$, and similarly for
$|T|$.) 
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II.  All parts of this question refer to the matrix
$$A = \pmatrix{1 & -2 & 0 &  s  & 2\cr
               -2 & -1 & -1 & 2s & -1\cr
                1 & 3 & 1 & s^2 & -1\cr}$$
\item{A)} (15) For which value(s) of the scalar $s$ does $A$ satisfy
$\dim {\rm Nul}(A) = 2$?  
\item{B)} (10) Say $s = 1$.  Give bases of ${\rm Col}(A)$ and 
${\rm Nul}(A)$.
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III.  (10)  Use Cramer's Rule to solve the system
$$\eqalign{3x_1 + 2 x_2 &= 9\cr
          -8x_1 + 3 x_2 &= -12\cr}$$
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IV.  Let $V = M_{2\times 2}({\bf R})$, the vector space of all 
$2\times 2$ matrices with real entries.  Let $W = {\bf R}^2$.
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\item{A)} (10) Is $T: V \to W$ defined by $T(A) = \pmatrix{\det(A) \cr \det(A^2)\cr}$
a linear mapping?  Why or why not?
\item{B)} (10) Show that
$$H = \left\{A = \pmatrix{a & b\cr c & d\cr} \in V : a + b = c + d\right\}$$
is a vector subspace of $V$.  
\item{C)} (5 {\bf Extra Credit})  Find a basis for $H$ from part B and 
determine its dimension.
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\noindent
V.  (True-False)  For each true statement, give a short proof.
For each false statement, give a counterexample.  (Note:
As always, ``true'' means ``true in every case.'' Partial credit {\it will}
be given for definitions or theorems that apply even if you don't find a 
complete solution.)
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\item{A)} (10) Let $A$ be a $4\times 4$ matrix in which 
$R_1 + 2 R_2 = R_3 + R_4$  ($R_i$ is row $i$ of the matrix). 
Then $\det(A) = 0$.  
\item{B)} (10) There exist linear mappings
$T : {\bf R}^2 \to {\bf R}^2$ with standard matrices $A$ satisfying
${\rm Nul}(A) = {\rm Col}(A)$.
\item{C)} (10) Let $A,B,C$ be $n \times n$ matrices.  If $\det(A^t B C^5) = 23$,
then $A,B,C$ are {\it all} invertible matrices.  
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