Linear Algebra with Applications, 10th Edition

5.6

1. 1. 

   1. (a) $\left\{(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}{)}^T,(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}{)}^T\right\}$;

   2. (b) $\left\{(\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}}{)}^T,(-\frac{1}{\sqrt{5}},\frac{2}{\sqrt{5}}{)}^T\right\}$

2. 2. 

   1. (a) $\left[\begin{array}{cc}\hfill -\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\ \hfill \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{cc}\sqrt{2}& \hfill \sqrt{2}\\ 0& \hfill 4\sqrt{2}\end{array}\right]$;

   2. (b) $\left[\begin{array}{cc}\hfill \frac{2}{\sqrt{5}}& \hfill -\frac{1}{\sqrt{5}}\\ \frac{1}{\sqrt{5}}& \hfill \frac{2}{\sqrt{5}}\end{array}\right]\left[\begin{array}{cc}\sqrt{5}& \hfill 4\sqrt{5}\\ 0& \hfill 3\sqrt{5}\end{array}\right]$

3. 3. $\{{(\frac{1}{3},\frac{2}{3},-\frac{2}{3})}^T,{(\frac{2}{3},\frac{1}{3},\frac{2}{3})}^T,{(-\frac{2}{3},\frac{2}{3},\frac{1}{3})}^T\}$

4. 4. $u_1(x)=\frac{1}{\sqrt{2}},u_2(x)=\frac{\sqrt{6}}{2}x,u_3(x)=\frac{3\sqrt{10}}{4}(x^2-\frac{1}{3})$

5. 5. 

   1. (a) $\left\{\frac{1}{3}(2,1,2{)}^T,\frac{\sqrt{2}}{6}(-1,4,-1{)}^T\right\}$;

   2. (b) $\begin{array}{cc}Q=\left[\begin{array}{cc}\frac{2}{3}& \hfill \frac{-\sqrt{2}}{6}\\ \frac{1}{3}& \hfill \frac{2\sqrt{2}}{3}\\ \frac{2}{3}& \hfill \frac{-\sqrt{2}}{6}\end{array}\right];& R=\left[\begin{array}{cc}3& \hfill \frac{5}{3}\\ 0& \hfill \frac{\sqrt{2}}{3}\end{array}\right]\end{array}$;

   3. (c) $\mathbf{x}=\left[\begin{array}{r}9\\ -3\end{array}\right]$

6. 6. 

   1. (b) $\left[\begin{array}{cc}\frac{3}{5}& \hfill -\frac{4}{5\sqrt{2}}\\ \frac{4}{5}& \hfill \frac{3}{5\sqrt{2}}\\ 0& \hfill \frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}{cc}5& 1\\ 0& 2\sqrt{2}\end{array}\right]$;

   2. (c) ${(2.1,5.5)}^T$

7. 7. $\{{(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0,0)}^T,{(\frac{\sqrt{2}}{3},\frac{\sqrt{2}}{3},-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{6})}^T\}$

8. 8. $\left\{\right[\begin{array}{c}\frac{4}{5}\\ \frac{2}{5}\\ \frac{2}{5}\\ \frac{1}{5}\end{array}],[\begin{array}{r}\frac{1}{5}\\ -\frac{2}{5}\\ -\frac{2}{5}\\ \frac{4}{5}\end{array}],[\begin{array}{r}0\\ \frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\\ 0\end{array}\left]\right\}$

9. 13.  

   1. (a) Hint: Make use of Theorems 5.5.7 and 5.5.8.

   2. (b) Hint: Use the result from part (a).

   3. (c) Hint: The projection matrix corresponding to a subspace is unique.

10. 15. Hint: $\text{dim}(U\cap V)=r>0$, start with an orthonormal basis $\{{\mathbf{\text{v}}}_1,{\mathbf{\text{v}}}_2,\mathrm{...},{\mathbf{\text{v}}}_r\}$ for $U\cap V$ and extend it to an orthonormal basis for $\{{\mathbf{\text{v}}}_1,{\mathbf{\text{v}}}_2,\mathrm{...},{\mathbf{\text{v}}}_k\}$ for V. Let $W={\text{Span}(\mathbf{\text{v}}}_{r+1},\mathrm{...}$