Linear Algebra with Applications, 10th Edition

Section 5.5 Exercises

1. Which of the following sets of vectors form an orthonormal basis for ${ℝ}^2$?

   1. $\{{(1,0)}^T,{(0,1)}^T\}$

   2. $\left\{{\left({\displaystyle \frac{3}{5}},{\displaystyle \frac{4}{5}}\right)}^T,{\left({\displaystyle \frac{5}{13}},{\displaystyle \frac{12}{13}}\right)}^T\right\}$

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

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

2. Let

   $${\mathbf{\text{u}}}_1=\left[\begin{array}{c}\hfill \frac{1}{3\sqrt{2}}\\ \hfill \frac{1}{3\sqrt{2}}\\ \hfill -\frac{4}{3\sqrt{2}}\end{array}\right],{\mathbf{\text{u}}}_2=\left[\begin{array}{c}\hfill \frac{2}{3}\\ \hfill \frac{2}{3}\\ \hfill \frac{1}{3}\end{array}\right],{\mathbf{\text{u}}}_3=\left[\begin{array}{c}\hfill \frac{1}{2}\\ \hfill -\frac{1}{\sqrt{2}}\\ \hfill 0\end{array}\right]$$

   1. Show that $\{{\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{u}}_3\}$ is an orthonormal basis for ${ℝ}^2$.

   2. Let $\mathbf{x}={(1,1,)}^T$. Write x as a linear combination of ${\mathbf{u}}_1,{\mathbf{u}}_2$, and ${\mathbf{u}}_3$ using Theorem 5.5.2 and use Parseval’s formula to compute $\Vert \mathbf{x}\Vert$.

3. Let S be the subspace of ${ℝ}^3$ spanned by the vectors ${\mathbf{u}}_2$ and ${\mathbf{u}}_3$ of Exercise 2. Let $\mathbf{x}={(1,2,2)}^T$. Find the projection p of x onto S. Show that $(\mathbf{p}-\mathbf{x})\perp {\mathbf{u}}_2$ and $(\mathbf{p}-\mathbf{x})\perp {\mathbf{u}}_3$.

4. Let θ be a fixed real number and let

   $$\begin{array}{cc}\hfill {\mathbf{x}}_1=\left[\begin{array}{c}\hfill \cos \theta \\ \hfill \sin \theta \end{array}\right]& \hfill \begin{array}{cc}\hfill \text{and}& \hfill {\mathbf{x}}_2=\left[\begin{array}{c}\hfill -\sin \theta \\ \hfill \cos \theta \end{array}\right]\end{array}\end{array}$$

   1. Show that $\{{\mathbf{x}}_1,{\mathbf{x}}_2\}$ is an orthonormal basis for ${ℝ}^2$.

   2. Given a vector y in ${ℝ}^2$, write it as a linear combination ...