Linear Algebra with Applications, 10th Edition

Chapter 5 Exercises

MATLAB Exercises

1. Set

   $$\begin{array}{ccc}\mathbf{x}=[0:4,4,-4,1,1]\prime & \text{and}& \mathsf{\text{y}}=\mathsf{\text{ones}}(9,1)\end{array}$$

   1. Use the MATLAB function norm to compute the values of $\parallel \mathbf{x}\parallel ,\parallel \mathbf{y}\parallel ,\parallel \mathbf{x}+\mathbf{y}\parallel$ and to verify that the triangle inequality holds. Use MATLAB also to verify that the parallelogram law

      $$\parallel \mathbf{\text{x+y}}{\parallel }^2+\parallel \mathbf{x}-\mathbf{y}{\parallel }^2=2(\parallel \mathbf{x}{\parallel }^2+\parallel \mathbf{y}{\parallel }^2)$$

      is satisfied.

   2. If

      $$t=\frac{{\displaystyle {\mathbf{x}}^T\mathbf{y}}}{{\displaystyle \parallel \mathbf{x}\parallel \parallel \mathbf{y}\parallel }}$$

      then why do we know that $\left|t\right|$ must be less than or equal to 1? Use MATLAB to compute the value of t and use the MATLAB function acos to compute the angle between x and y. Convert the angle to degrees by multiplying by $180/\pi$. (Note that the number $\pi$ is given by pi in MATLAB.)

   3. Use MATLAB to compute the vector projection p of x onto y. Set $\mathbf{z}=\mathbf{x}-\mathbf{p}$ and verify that z is orthogonal to p by computing the scalar product of the two vectors. Compute ...