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\textbf{Math 228: Intro to Lin Alg \& Diff Eqns}
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\textbf{Quiz 6}\quad \textsl{March 7, 2002}
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1. Let 
$B= \left\{ (1, 1, 0), \,\,\,  (1, 0, 1), \,\,\, (0, 1, 1) \right\}$
and 
$C= \left\{ (1, 0, -1), \,\,\, (1, 0, 0), \,\,\, (0, -1, 1) \right\}$,
two bases of $\R^{3}$. Find the transition matrix from $B$ to $C$.

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2. Let $T(x,y,z) = (3x-2y, 4z-x, 2x-y, 4z-y).$ Find a matrix $A$
such that $T(\x) = A\x$.

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3. If
$U=\left[\begin{array}{cc}
		u_{1} & u_{2} \\
		0 & u_{3} 
		\end{array} \right]$ and
$V=\left[ \begin{array}{cc}
		v_{1} & v_{2} \\
		0 & v_{3} 
		\end{array} \right]$,
define $U \cdot V = u_{1}v_{1} + u_{3}v_{3}.$
Determine whether this formula defines an inner product on the space of all
$2 \times 2$ upper triangular matrices.


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