Section 6.7

1.
1. (a) F
2. (b) F
3. (c) T
4. (d) T
5. (e) F
6. (f) F
7. (g) T
2.
1. (a) $\begin{array}{l} v_{1} = (\begin{matrix} 1 \\ 0 \end{matrix}), v_{2} = (\begin{matrix} 0 \\ 1 \end{matrix}), u_{1} = \frac{1}{\sqrt{3}} (\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}), u_{2} = \frac{1}{\sqrt{2}} (\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}), u_{3} = \frac{1}{\sqrt{6}} (\begin{array}{r} 2 \\ - 1 \\ - 1 \end{array}) \\ σ_{1} = \sqrt{3}, σ_{2} = \sqrt{2} \end{array}$ $\begin{array}{l} v_{1} = (\begin{matrix} 1 \\ 0 \end{matrix}), v_{2} = (\begin{matrix} 0 \\ 1 \end{matrix}), u_{1} = \frac{1}{\sqrt{3}} (\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}), u_{2} = \frac{1}{\sqrt{2}} (\begin{array}{r} 0 \\ 1 \\ - 1 \end{array}), u_{3} = \frac{1}{\sqrt{6}} (\begin{array}{r} 2 \\ - 1 \\ - 1 \end{array}) \\ σ_{1} = \sqrt{3}, σ_{2} = \sqrt{2} \end{array}$
2. (c) $\begin{array}{l} v_{1} = \frac{1}{\sqrt{π}} \sin x, v_{2} = \frac{1}{\sqrt{π}} \cos x, v_{3} = \frac{1}{\sqrt{2 π}} \\ u_{1} = \frac{\cos x + 2 \sin x}{\sqrt{5 π}}, u_{2} = \frac{2 \cos x - \sin x}{\sqrt{5 π}}, u_{3} = \frac{1}{\sqrt{2 π}}, \\ σ_{1} = \sqrt{5}, σ_{2} = \sqrt{5}, σ_{3} = 2 \end{array}$ $\begin{array}{l} v_{1} = \frac{1}{\sqrt{π}} \sin x, v_{2} = \frac{1}{\sqrt{π}} \cos x, v_{3} = \frac{1}{\sqrt{2 π}} \\ u_{1} = \frac{\cos x + 2 \sin x}{\sqrt{5 π}}, u_{2} = \frac{2 \cos x - \sin x}{\sqrt{5 π}}, u_{3} = \frac{1}{\sqrt{2 π}}, \\ σ_{1} = \sqrt{5}, σ_{2} = \sqrt{5}, σ_{3} = 2 \end{array}$
3.
1. (a) $(\begin{array}{r} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ - \frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{6}} \end{array}) (\begin{array}{r} \sqrt{6} & 0 \\ 0 & 0 \\ 0 & 0 \end{array}) {(\begin{array}{r} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{array})}^{*}$ $(\begin{array}{r} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ \frac{1}{\sqrt{3}} & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\ - \frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{6}} \end{array}) (\begin{array}{r} \sqrt{6} & 0 \\ 0 & 0 \\ 0 & 0 \end{array}) {(\begin{array}{r} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{array})}^{*}$
2. (c) $(\begin{array}{r} \frac{2}{\sqrt{10}} & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{10}} \\ \frac{1}{\sqrt{10}} & - \frac{1}{\sqrt{2}} & 0 & - \frac{2}{\sqrt{10}} \\ \frac{1}{\sqrt{10}} & \frac{1}{\sqrt{2}} & 0 & - \frac{2}{\sqrt{10}} \\ \frac{2}{\sqrt{10}} & 0 & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{10}} \end{array}) (\begin{array}{r} \sqrt{5} & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}) {(\begin{array}{r} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{array})}^{*}$ $(\begin{array}{r} \frac{2}{\sqrt{10}} & 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{10}} \\ \frac{1}{\sqrt{10}} & - \frac{1}{\sqrt{2}} & 0 & - \frac{2}{\sqrt{10}} \\ \frac{1}{\sqrt{10}} & \frac{1}{\sqrt{2}} & 0 & - \frac{2}{\sqrt{10}} \\ \frac{2}{\sqrt{10}} & 0 & - \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{10}} \end{array}) (\begin{array}{r} \sqrt{5} & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{array}) {(\begin{array}{r} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{array})}^{*}$
3. (e) $(\begin{array}{r} \frac{1 + i}{2} & \frac{1 + i}{2} \\ \frac{1 - i}{2} & \frac{- 1 + i}{2} \end{array}) (\begin{array}{r} \sqrt{6} & 0 \\ 0 & 0 \end{array}) {(\begin{array}{r} \frac{2}{\sqrt{6}} & \frac{1 - i}{\sqrt{6}} \\ \frac{1 + i}{\sqrt{6}} & - \frac{2}{\sqrt{6}} \end{array})}^{*}$ $(\begin{array}{r} \frac{1 + i}{2} & \frac{1 + i}{2} \\ \frac{1 - i}{2} & \frac{- 1 + i}{2} \end{array}) (\begin{array}{r} \sqrt{6} & 0 \\ 0 & 0 \end{array}) {(\begin{array}{r} \frac{2}{\sqrt{6}} & \frac{1 - i}{\sqrt{6}} \\ \frac{1 + i}{\sqrt{6}} & - \frac{2}{\sqrt{6}} \end{array})}^{*}$
4.
1. (a) $W P = (\begin{array}{r} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{array}) (\begin{array}{r} \frac{\sqrt{8} + \sqrt{2}}{2} & \frac{- \sqrt{8} + \sqrt{2}}{2} \\ \frac{- \sqrt{8} + \sqrt{2}}{2} & \frac{\sqrt{8} + \sqrt{2}}{2} \end{array})$ $W P = (\begin{array}{r} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{array}) (\begin{array}{r} \frac{\sqrt{8} + \sqrt{2}}{2} & \frac{- \sqrt{8} + \sqrt{2}}{2} \\ \frac{- \sqrt{8} + \sqrt{2}}{2} & \frac{\sqrt{8} + \sqrt{2}}{2} \end{array})$
5.
1. (a) $T^{†} (x, y, z) = (\frac{x + y + z}{3}, \frac{y - z}{2})$ $T^{†} (x, y, z) = (\frac{x + y + z}{3}, \frac{y - z}{2})$
2. (c) $T^{†} (a + b \sin x + c \cos x) = T^{- 1} (a + b \sin x + c \cos x) = \frac{a}{2} + \frac{(2 b + c) \sin x + (- b + 2 c) \cos x}{5}$ $T^{†} (a + b \sin x + c \cos x) = T^{- 1} (a + b \sin x + c \cos x) = \frac{a}{2} + \frac{(2 b + c) \sin x + (- b + 2 c) \cos x}{5}$
6.
1. (a) $\frac{1}{6} (\begin{array}{r} 1 & 1 & - 1 \\ 1 & 1 & - 1 \end{array})$ $\frac{1}{6} (\begin{array}{r} 1 & 1 & - 1 \\ 1 & 1 & - 1 \end{array})$
2. (c) $\frac{1}{5} (\begin{array}{r} 1 & - 2 & 3 & 1 \\ 1 & 3 & - 2 & 1 \end{array})$ $\frac{1}{5} (\begin{array}{r} 1 & - 2 & 3 & 1 \\ 1 & 3 & - 2 & 1 \end{array})$
3. (e) $\frac{1}{6} (\begin{array}{c} 1 - i & 1 + i \\ 1 & i \end{array})$ $\frac{1}{6} (\begin{array}{c} 1 - i & 1 + i \\ 1 & i \end{array})$
7.
1. (a) $Z_{1} = N {(T)}^{⊥} = R^{2} and Z_{2} = R (T) = span {(1, 1, 1), (0, 1, - 1)}$ $Z_{1} = N {(T)}^{⊥} = R^{2} and Z_{2} = R (T) = span {(1, 1, 1), (0, 1, - 1)}$
2. (c) $Z_{1} = N {(T)}^{⊥} = V and Z_{2} = R (T) = V$ $Z_{1} = N {(T)}^{⊥} = V and Z_{2} = R (T) = V$
8.
1. (a) No solution $\frac{1}{2} (\begin{array}{r} 1 \\ 1 \end{array})$ $\frac{1}{2} (\begin{array}{r} 1 \\ 1 \end{array})$

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Table of Contents for Section 6.7

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Table of Contents for
Section 6.7