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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Exam 1 -- February 16, 2007}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 points.
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I. 
\item{A)} (10)
For which value(s) of $c \in {\bf R}$ is the system of linear equations
represented
by the following augmented matrix {\it consistent}? 
$$\pmatrix{-1 & 2 & 0 & c &|&  1\cr
            0 & 1 & 2 & 0 &|& -1\cr
            2 & -4 & 0 & 8 &|&  5\cr}$$
\item{B)}  (10) Give a parametrization
of the set of solutions for a value of $c$ (your choice)
that makes the system consistent.
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II.  
\item{A)}  (5) {\it Define:}  The set $S$ of vectors in ${\bf R}^n$
is {\it linearly independent}.
\item{B)}  (10) For which value of $a \in {\bf R}$ is
$$S = \left\{\pmatrix{3\cr 2\cr 1 \cr},\pmatrix{-1\cr 1\cr 0\cr},
\pmatrix{1\cr 0\cr a\cr}\right\}$$
a linearly {\it dependent} subset of ${\bf R}^3$?  
\item{C)}  (15) Let $b = \pmatrix{3\cr 3\cr 4\cr}$.  Using your 
value of $a$ from part B, is $b \in {\rm Span}(S)$?  Why or why not?
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III.  Let $T : {\bf R}^3 \to {\bf R}^3$ be the mapping defined by
$$T(x) = \pmatrix{1 & 2 & 1\cr
                 -1 & 4 & 0\cr
                  1 & 0 & 1\cr}x$$
\item{A)} (10) Show that $T$ is a linear mapping by verifying the 
properties in the definition.
\item{B)} (10) Is $T$ an onto mapping?  Why or why not?  
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IV.  (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 the homogeneous system $Ax = 0$ has two free variables,
then the set of solutions of $Ax = b$ is a plane for all $b$. 
\item{B)} (10)  If $x \in {\rm Span}(S)$, then the set
$S \cup \{x\}$ is linearly dependent.  
\item{C)} (10)  Let $A$ be an $m\times n$ matrix that defines an {\it onto}
mapping $T_A : {\bf R}^n \to {\bf R}^m$ ($T_A(x) = Ax$ for $x \in {\bf R}^n$).
Then the span of the set of columns of $A$ is ${\bf R}^m$.  
\bye
