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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Midterm Exam 2 -- March 26, 2004}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  There are 100 regular points
in questions I - III, and 10 Extra Credit points.   Standard 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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\noindent
I.  Let $T : P_3({\bf R}) \to {\bf R}^3$ be defined 
by $$T(p(x)) = (p(1),p'(1),p''(1)).$$
\item{A)} (10)  Show that $T$ is linear.
\item{B)} (10)  Show that $\alpha = \{1,x+x^2,x^2+x^3,x^3+1\}$ 
is a basis for the vector space $P_3({\bf R})$ (don't work too hard!)
\item{C)} (10) Find the matrix of $T$ with respect to the
bases $\alpha = \{1,x+x^2,x^2+x^3,x^3+1\}$ for $P_3({\bf R})$ and
and $\beta = $ the standard basis in ${\bf R}^3$.
\item{D)} (15) Let $S: {\bf R}^3 \to {\bf R}^3$ 
be the linear mapping defined by 
$$S(x_1,x_2,x_3) = (3x_1-x_2,x_1+x_2+x_3,x_1).$$  
What is the
matrix of $ST : P_3({\bf R}) \to {\bf R}^3$, with respect
to the bases $\alpha$ and $\beta$ as in part C?
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\noindent
II. (25)  State and prove the Dimension Theorem for
linear mappings.
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\noindent
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 $T : {\bf R}^3 \to {\bf R}^5$, then
there are some ${\bf b} \in {\bf R}^5$ such that the
equation $T({\bf x}) = {\bf b}$ has no solutions ${\bf x}\in {\bf R}^3$.
\item{B)} (10)  If $T: P({\bf R}) \to P({\bf R})$ is a linear
mapping and $T$ is not injective, then $T$ is not surjective either.
\item{C)} (10) There exists a linear
mapping $T : {\bf R}^4 \to {\bf R}^7$ which has $\dim {\rm Ker}(T) = 2$,
and whose image contains the vectors 
$$(1,1,1,0,0,0,0), \quad (0,0,0,1,1,1,1), \quad (1,0,1,0,1,0,1).$$ 

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\noindent
{\it Extra Credit}  (10)  We know one proof of the following statement
via the process of reduction of a system of equations to echelon form:
{\it Every homogeneous system of $m$ linear equations in $n$ variables
with $n > m$ has a nontrivial solution.}  Give {\it a different}
proof of this statement using the Dimension Theorem for linear mappings 
(problem II).
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