Section 2.6

1.
1. (a) F
2. (b) T
3. (c) T
4. (d) T
5. (e) F
6. (f) T
7. (g) T
8. (h) F
2. The functions in (a), (c), (e), and (f) are linear functionals.
3. (a) $f_{1} (x, y, z) = x - \frac{1}{2} y, f_{2} (x, y, z) = \frac{1}{2} y,$ $f_{1} (x, y, z) = x - \frac{1}{2} y, f_{2} (x, y, z) = \frac{1}{2} y,$ and $f_{3} (x, y, z) = - x + z$ $f_{3} (x, y, z) = - x + z$
5. The basis for V is ${p_{1} (x), p_{2} (x)},$ ${p_{1} (x), p_{2} (x)},$ where $p_{1} (x) = 2 - 2 x$ $p_{1} (x) = 2 - 2 x$ and $p_{2} (x) = - \frac{1}{2} + x$ $p_{2} (x) = - \frac{1}{2} + x$ .
7.
1. (a) $T^{t} (f) = g,$ $T^{t} (f) = g,$ where $g (a + b x) = - 3 a - 4 b$ $g (a + b x) = - 3 a - 4 b$
2. (b) ${[T^{t}]}_{γ *}^{β *} = (\begin{array}{l} - 1 & 1 \\ - 2 & 1 \end{array})$ ${[T^{t}]}_{γ *}^{β *} = (\begin{array}{l} - 1 & 1 \\ - 2 & 1 \end{array})$
3. (c) ${[T]}_{β}^{γ} = (\begin{array}{r} - 1 & - 2 \\ 1 & 1 \end{array})$ ${[T]}_{β}^{γ} = (\begin{array}{r} - 1 & - 2 \\ 1 & 1 \end{array})$

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Table of Contents for Section 2.6