Section 2.5

1.
1. (a) F
2. (b) T
3. (c) T
4. (d) F
5. (e) T
2.
1. (a) $(\begin{array}{c} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array})$ $(\begin{array}{c} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array})$
2. (c) $(\begin{array}{l} 3 & - 1 \\ 5 & - 2 \end{array})$ $(\begin{array}{l} 3 & - 1 \\ 5 & - 2 \end{array})$
3.
1. (a) $(\begin{array}{c} a_{2} & b_{2} & c_{2} \\ a_{1} & b_{1} & c_{1} \\ a_{0} & b_{0} & c_{0} \end{array})$ $(\begin{array}{c} a_{2} & b_{2} & c_{2} \\ a_{1} & b_{1} & c_{1} \\ a_{0} & b_{0} & c_{0} \end{array})$
2. (c) $(\begin{array}{r} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ - 3 & 2 & 1 \end{array})$ $(\begin{array}{r} 0 & - 1 & 0 \\ 1 & 0 & 0 \\ - 3 & 2 & 1 \end{array})$
3. (e) $(\begin{array}{r} 5 & - 6 & 3 \\ 0 & 4 & - 1 \\ 3 & - 1 & 2 \end{array})$ $(\begin{array}{r} 5 & - 6 & 3 \\ 0 & 4 & - 1 \\ 3 & - 1 & 2 \end{array})$
4. ${[T]}_{β^{'}} = (\begin{array}{r} 2 & - 1 \\ - 1 & 1 \end{array}) (\begin{array}{r} 2 & 1 \\ 1 & - 3 \end{array}) (\begin{array}{r} 1 & 1 \\ 1 & 2 \end{array}) = (\begin{array}{r} 8 & 13 \\ - 5 & - 9 \end{array})$ ${[T]}_{β^{'}} = (\begin{array}{r} 2 & - 1 \\ - 1 & 1 \end{array}) (\begin{array}{r} 2 & 1 \\ 1 & - 3 \end{array}) (\begin{array}{r} 1 & 1 \\ 1 & 2 \end{array}) = (\begin{array}{r} 8 & 13 \\ - 5 & - 9 \end{array})$
5. ${[T]}_{β^{'}} = (\begin{array}{r} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{2} \end{array}) (\begin{array}{r} 0 & 1 \\ 0 & 0 \end{array}) (\begin{array}{r} 1 & 1 \\ 1 & - 1 \end{array}) = (\begin{array}{r} \frac{1}{2} & - \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{2} \end{array})$ ${[T]}_{β^{'}} = (\begin{array}{r} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{2} \end{array}) (\begin{array}{r} 0 & 1 \\ 0 & 0 \end{array}) (\begin{array}{r} 1 & 1 \\ 1 & - 1 \end{array}) = (\begin{array}{r} \frac{1}{2} & - \frac{1}{2} \\ \frac{1}{2} & - \frac{1}{2} \end{array})$
6.
1. (a) $Q = (\begin{array}{r} 1 & 1 \\ 1 & 2 \end{array}), {[L_{A}]}_{β} = (\begin{array}{r} 6 & 11 \\ - 2 & - 4 \end{array})$ $Q = (\begin{array}{r} 1 & 1 \\ 1 & 2 \end{array}), {[L_{A}]}_{β} = (\begin{array}{r} 6 & 11 \\ - 2 & - 4 \end{array})$
2. (c) $Q = (\begin{array}{r} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{array}), {[L_{A}]}_{β} = (\begin{array}{r} 2 & 2 & 2 \\ - 2 & - 3 & - 4 \\ 1 & 1 & 2 \end{array})$ $Q = (\begin{array}{r} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{array}), {[L_{A}]}_{β} = (\begin{array}{r} 2 & 2 & 2 \\ - 2 & - 3 & - 4 \\ 1 & 1 & 2 \end{array})$
7. (a) $T (x, y) = \frac{1}{1 + m^{2}} ((1 - m^{2}) x + 2 m y, 2 m x + (m^{2} - 1) y)$ $T (x, y) = \frac{1}{1 + m^{2}} ((1 - m^{2}) x + 2 m y, 2 m x + (m^{2} - 1) y)$

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Table of Contents for Section 2.5

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Table of Contents for
Section 2.5