1.

1. (a) F

2. (b) T

3. (c) T

4. (d) F

5. (e) T

6. (f) F

7. (g) T

2. For each part the orthonormal basis and the Fourier coefficients are given.

1. (b) $\left\{{\displaystyle \frac{\sqrt{3}}{3}}(1,1,1),{\displaystyle \frac{\sqrt{6}}{6}}(-2,1,1),{\displaystyle \frac{\sqrt{2}}{2}}(0,-1,1)\right\};{\displaystyle \frac{2\sqrt{3}}{3}},-{\displaystyle \frac{\sqrt{6}}{6}},{\displaystyle \frac{\sqrt{2}}{2}}.$

2. (c) $\left\{1,2\sqrt{3}(x-{\displaystyle \frac{1}{2}}),6\sqrt{5}(x^2-x+{\displaystyle \frac{1}{6}})\right\};{\displaystyle \frac{3}{2}},{\displaystyle \frac{\sqrt{3}}{6}},0.$

3. (e) $\left\{{\displaystyle \frac{1}{5}}(2,-1,-2,4),{\displaystyle \frac{1}{\sqrt{30}}}(-4,2,-3,1),{\displaystyle \frac{1}{\sqrt{155}}}(-3,4,9,7)\right\};10,3\sqrt{30},\sqrt{155}$

4. (g) $\left\{{\displaystyle \frac{1}{6}}\left(\begin{array}{rr}3& 5\\ -1& 1\end{array}\right),{\displaystyle \frac{1}{6\sqrt{2}}}\left(\begin{array}{rr}-4& 4\\ 6& -2\end{array}\right),{\displaystyle \frac{1}{9\sqrt{2}}}\left(\begin{array}{rr}9& -3\\ 6& -6\end{array}\right)\right\};24,6\sqrt{2},-9\sqrt{2}$

5. (i) $\begin{array}{l}\left\{\sqrt{{\displaystyle \frac{2}{\pi }}}\sin t,\sqrt{{\displaystyle \frac{2}{\pi }}}\cos t,\sqrt{{\displaystyle \frac{\pi }{{\pi }^2-8}}}(1-{\displaystyle \frac{4}{\pi }}\sin t),\sqrt{{\displaystyle \frac{12\pi }{{\pi }^4-96}}}(t+{\displaystyle \frac{4}{\pi }}\cos t-{\displaystyle \frac{\pi }{2}})\right\};\\ \sqrt{{\displaystyle \frac{2}{\pi }}}(2\pi +2),-4\sqrt{{\displaystyle \frac{2}{\pi }}},\sqrt{{\displaystyle \frac{{\pi }^2-8}{\pi }}}(1+\pi ),\sqrt{{\displaystyle \frac{{\pi }^4-96}{3\pi }}}\end{array}$

6. (k) $\begin{array}{l}\{{\displaystyle \frac{1}{\sqrt{47}}}(-4,3,-2i,i,1-4i),{\displaystyle \frac{1}{\sqrt{60}}}(3-i,-5i,-2+4i,2+i),\\ {\displaystyle \frac{1}{\sqrt{1160}}}(-17-i,-9+8i,-18+16i,-9+8i)\};\\ \sqrt{47}(-1-i),\sqrt{60}(-1+2i),\sqrt{1160}(1+i)\end{array}$

7. (m) $\begin{array}{l}\{{\displaystyle \frac{1}{\sqrt{18}}}\left(\begin{array}{cc}-1+i& -i\\ 2-i& 1+3i\end{array}\right),{\displaystyle \frac{1}{\sqrt{246}}}\left(\begin{array}{cc}-4i& -11-9i\\ 1+5i& 1-i\end{array}\right),\\ {\displaystyle \frac{1}{\sqrt{39063}}}\left(\begin{array}{cc}-5-118i& -7-26i\\ -145i& -58\end{array}\right)\};\sqrt{18}(2+i),\sqrt{246}(-1-i),0\end{array}$

4. $S^{\perp }=\text{span}(\{(i,-{\displaystyle \frac{1}{2}}(1+i),1)\})$

5. ${{\displaystyle S}}_0^{\perp }$ is the plane through the origin that is perpendicular to $x_0;S^{\perp }$ is the line through the origin that is perpendicular to the plane containing $x_1$and $x_2$

19.

1. (a) ${\displaystyle \frac{1}{17}}\left(\begin{array}{r}26\\ 104\end{array}\right)$

2. (b) ${\displaystyle \frac{1}{14}}\left(\begin{array}{c}29\\ 17\\ 40\end{array}\right)$

20.

1. (b) ${\displaystyle \frac{1}{\sqrt{14}}}$