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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Sample Exam 2 -- March 16, 2007}
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\noindent
I.  All parts of this question refer to the matrix
$$A = \pmatrix{1 & 0 & 2 & 3 & -1\cr
               2 & 1 & -1 & 1 & s\cr
              -1 &-1 & 3 & 2 & -1\cr}$$
\item{A)}  For which value of the scalar $s$ does $A$ satisfy
${\rm rank}(A) = 2$?  In the next two parts use this value of $s$.
\item{B)}  Give a basis of ${\rm Col}(A)$.
\item{C)}  Find the dimension of ${\rm Nul}(A)$ and give a basis of ${\rm Nul}(A)$.
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\noindent
II.  
\item{A)}  Use Cramer's Rule to solve
$$\eqalign{
5 x_1 + 3x_2 - x_3 &= 1\cr
3 x_1 + x_2 + x_3  &= 2\cr
x_1 + x_2 - x_3 &= 0\cr}$$
\item{B)}  Prove Cramer's Rule:  If $A$ is an invertible $n\times n$ matrix,
then the solution of $Ax = b$ is the vector $x$ with entries
$$x_i = {\det(A_i(b))\over \det(A)}$$
for $i = 1, \ldots, n$.  
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\noindent
III.  Let $V = P_3({\bf R})$, the vector space of all polynomials
of degree $\le 3$ with real coefficients.
\item{A)}  Is $T: V \to V$ defined by $T(p) = p' - 2p$ a linear 
transformation ($p'$ = derivative of $p$)?  Why or why not?
\item{B)}  Is 
$$W = \{p \in V : p'(x) - 2p(x) = 0 {\rm\ for\ all\ } x\}$$ 
a vector subspace of $V$?
Why or why not? 
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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.'')
\item{A)} If $T : {\bf R}^5 \to {\bf R}^3$ is linear, $A$
is the standard matrix of $T$, 
and $S = \{v_1,v_2,v_3\} \subset {\rm Nul}(A)$, then
$S$ is linearly dependent.
\item{B)} Let $A,B$ be $n\times n$ matrices.
If $\det(A) = -3$ and $B^2 = I$, then $B^t A B$ is an invertible matrix.  
\item{C)}  If $A$ is an $m\times n$ matrix, then 
${\rm rank}(A) + \dim {\rm Nul}(A) = n$.  
\bye
