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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Sample Exam 1 -- February 9, 2007}
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{\it Directions}
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There are 100 points.
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I.  Let $S$ be a subset of ${\bf R}^n$.
\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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\noindent
II.  (15) Is 
$$\left\{\pmatrix{1\cr 1\cr 0\cr},\pmatrix{-1\cr 0\cr 2\cr},\pmatrix{0\cr 2\cr 1\cr}\right\}$$
a linearly independent subset of ${\bf R}^3$?  Why or why not?
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\noindent
III.  (15) Put the augmented matrix of this system of linear equations
into row-reduced 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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IV.  
\item{A)}  (5) Let $T : {\bf R}^3 \to {\bf R}^3$ be the linear mapping
with $T(e_1) = 5e_1 + e_3$, $T(e_2) = 2e_2 - e_3$, $T(e_3) = e_1 - e_2$.
What is the matrix $A$ such that $T = T_A$?
\item{B)} (15) Is the matrix $A$ you found in part A invertible?  If so,
find $A^{-1}$.  If not, give a reason why you know it is not invertible.
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\noindent
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)  Let $A$ be an $n\times n$
matrix.  If the system of linear equations $Ax = b$ has just one
solution $x$ for some $b \in {\bf R}^n$, then it has just one solution for all $b$.
\item{B)} (10)  If
$W = {\rm Span}(\{{\bf v}, {\bf w}\})$ 
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.
\bye
