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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Retest Exam 2 -- March 28, 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.  All parts of this question refer to the matrix
$$A = \pmatrix{ 1 & 4 & 2 & 1 & 2t\cr
               -3 & 2 & 2 & 3 & t^2\cr
                2 & 1 & 0 & -1 & -t\cr}$$
\item{A)} (15)  What is $\dim{\rm Nul}(A)$ if $t = 2$?  Find a 
basis for ${\rm Nul}(A)$ in this case. 
\item{B)} (10)  What is $\dim{\rm Col}(A)$ if $t = 4$.  Find
a basis for ${\rm Col}(A)$ in this case.   
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\noindent
II.  
\item{A)} (15) Give the general proof of Cramer's Rule
for systems $Ax = b$ with $A \in M_{n\times n}$ an invertible
matrix.
\item{B)} (10) Give an example of a system $Ax = b$, where
$A$ is an invertible $3 \times 3$ matrix, but the second component
of the solution vector $x$ is $x_2 = 0$.
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III.  Let $V = P_2({\bf R})$, the vector space of polynomials
of degree $\le 2$, and let $W = M_{2\times 2}({\bf R})$, the vector space of all 
$2\times 2$ matrices with real entries. 
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\item{A)} (10) Show that $T: V \to W$ defined by 
$T(p) = \pmatrix{p(1) & p(2) \cr p'(1) & p'(2)\cr}$
($p' = $ derivative of $p$) is a linear mapping.  
\item{B)} (10) Let
$K = \left\{A = \pmatrix{a & b \cr c & d \cr} \in W\ |\ \dim {\rm Col}(A) \le 1\right\}$
Is $K$ a vector subspace of $W$?  Why or why not?
\item{C)} (5 {\bf Extra Credit})  Let $T$ be the mapping from 
part A.   What is ${\rm Ker}(T)$?  Justify your answer.
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\noindent
IV.  (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) If $a,b,c$ are distinct real numbers, then the matrix
$V = \pmatrix{1 & a & a^2\cr
           1 & b & b^2\cr
           1 & c & c^2\cr}$ 
is invertible.
\item{B)} (10)  If $A$ is a $5\times 7$ matrix,
then $\dim {\rm Nul} (A) \cdot \dim {\rm Col}(A) $ can equal 9.
\item{C)} (10)  Let 
$$A = \pmatrix{1 & -2 & 0 \cr
               -2 & -1 & -1 \cr}$$
Any $3\times 2$ matrix $X$ such that $AX = 0_{2\times 2}$
has linearly dependent columns.  
\bye
