Linear Algebra with Applications, 10th Edition

Section 5.4 Exercises

1. Let $\mathbf{x}={\left(-1,-1,1,1\right)}^T$ and $\mathbf{y}={\left(1,1,5,-3\right)}^T$. Show that $\mathbf{x}\perp \mathbf{y}$. Calculate ${\Vert \mathbf{\text{x}}\Vert }_2,{\Vert \mathbf{y}\Vert }_2,{\Vert \mathbf{x}+\mathbf{y}\Vert }_2$ and verify that the Pythagorean law holds.

2. Let $\mathbf{x}={\left(1,1,1,1\right)}^T$ and $\mathbf{y}={\left(8,2,2,0\right)}^T$.

   1. Determine the angle $\theta$ between x and y.

   2. Find the vector projection p of x onto y.

   3. Verify that $\mathbf{x}-\mathbf{p}$ is orthogonal to p.

   4. Compute ${\Vert \mathbf{x}-\mathbf{p}\Vert }_2,{\Vert \mathbf{p}\Vert }_2,{\Vert \mathbf{x}\Vert }_2$ and verify that the Pythagorean law is satisfied.

3. Use equation (1) with weight vector $\mathbf{w}={\left(\frac{1}{4},\frac{1}{2},\frac{1}{4}\right)}^T$ to define an inner product for ${ℝ}^3$, and let $\mathbf{x}={\left(1,1,1\right)}^T$ and $\mathbf{y}={\left(-5,1,3\right)}^T$.

   1. Show that x and y are orthogonal with respect to this inner product.

   2. Compute the values of $\Vert \mathbf{x}\Vert$ and $\Vert \mathbf{y}\Vert$ with respect to this inner product.

4. Given

   $$A=\left[\begin{array}{ccc}1& 2& 2\\ 1& 0& 2\\ 3& 1& 1\end{array}\right]\text{ and }B=\left[\begin{array}{ccc}-4& 1& 1\\ -3& 3& 2\\ 1& -2& -2\end{array}\right]$$

   determine the value of each of the following:

   1. $\langle A,B\rangle$

   2. ${\Vert A\Vert }_F$

   3. ${\Vert B\Vert }_{}$