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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Exam 1 -- February 20, 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.
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I.  (15) Let $V = P_4({\bf R})$, and
$$W = \{p\in V : p'(2) = 0\ {\rm and}\ p(0) = 0\}$$
Is $W$ a vector subspace of $V$?  Why or why not?
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II.  Let $S$ be a subset of a vector space $V$.
\item{A)} (5) Give the definition of ${\rm Span}(S)$, the 
{\it span\/} of $S$.
\item{B)}  (15) Show that $S$ is linearly dependent if and only if there
exists some ${\bf x} \in S$ which satisfies
${\bf x} \in {\rm Span}(S\ \backslash\ \{{\bf x}\})$.
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III.  (15) Is $\{(1,1,0),(-1,0,2),(0,2,1)\}$
a linearly independent subset of ${\bf R}^3$?  Why or why not?
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IV.  (20) Put this system of linear equations
into echelon form and find a parametrization for the set of solutions
$$\matrix{ 2x_1 &   &    & + & x_3 & - & x_4 & = &0 \cr
            x_1 & + &3x_2&   &     & + & 2x_4& = &1\cr
                &   & x_2& + & x_3 & + & x_4 & = &-1\cr}$$
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V.  (True-False)  Determine whether each of the following statements
is true or false.  For the ones that are true, give short proofs;
for those that are false, give counterexamples.
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\item{A)} (10)  If $S$ is linearly dependent in $V$ and 
$S \subset T$, then $T$ is also linearly dependent.
\item{B)} (10)  If $W$ is a subspace of $V$ and
$W = {\rm Span}(\{{\bf v}, {\bf w}\})$ 
then for any scalar $a \ne 0$, 
$W= {\rm Span}(\{a{\bf v} + {\bf w}, {\bf w}\})$ also.
\item{C)} (10)  If a system of linear equations has more variables
than unknowns, there is automatically a nonzero solution.
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{\it Extra Credit:}  (10)  In $F({\bf R})$, for each $a \in {\bf R}$,
we can define a function $f_a$ by 
$$f_a(x) = \cases{1 & if $x = a$\cr
                  0 & otherwise\cr}$$
Show that $S = \{f_a : a \in {\bf R}\}$ is a linearly independent
subset of $F({\bf R})$.  (For those who have taken Principles
of Analysis, note that this shows there is an {\it uncountably infinite}
linearly independent subset of ${\bf F}(R)$. It can be shown
that this implies that no basis can be even countably infinite.)
\bye