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\centerline{Mathematics 244, section 1 -- Linear Algebra}
\centerline{Makeup Exam 1 -- February 21, 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 inconsistent}? 
$$\pmatrix{-1 & 0 & 3 & c &|&  -1\cr
            0 & 1 & 2 & 0 &|&  0\cr
            3 & 0 & -9 & 12 &|&  2\cr}$$
\item{B)}  (10) Give a parametrization
of the set of solutions for a value of $c$ (your choice)
that makes the system {\it consistent}.
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\noindent
II.  
\item{A)}  (5) {\it Define:}  The set $S$ of vectors in ${\bf R}^n$
is {\it linearly dependent}.
\item{B)}  (10) For which value(s) of $a \in {\bf R}$ is
$$S = \left\{\pmatrix{1\cr 3\cr 1 \cr},\pmatrix{2\cr 4\cr 1\cr},
\pmatrix{-1\cr a\cr 1\cr}\right\}$$
a linearly {\it independent} subset of ${\bf R}^3$?  
\item{C)}  (15) Let $b = \pmatrix{3\cr 3\cr 4\cr}$.  Using
$a = 1$ in 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 linear
mapping with $T(e_1) = e_1 + e_3$, $T(e_2) = -e_2 + 2e_3$, $T(e_3) = e_1 + e_2$.  
\item{A)} (10) What is $T(4e_1 + 2e_2 - e_3)$?
\item{B)} (10) Is $T$ a one-to-one mapping?  Why or why not?  
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\noindent
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,
and the inhomogeneous system $Ax = b$ is consistent, then 
the set of solutions of $Ax = b$ is a plane. 
\item{B)} (10)  If $x \notin S$ and
$S \cup \{x\}$ is linearly dependent, then $x \in {\rm Span}(S)$.
\item{C)} (10) If $A$ is 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 $m \ge n$.  
\bye
