Linear Algebra with Applications, 10th Edition

Chapter 4 Exercises

MATLAB Exercises

1. Use MATLAB to generate a matrix W and a vector x by setting

   $$\begin{array}{cc}W=\mathsf{\text{t}}\mathsf{\text{r}}\mathsf{\text{i}}\mathsf{\text{u}}\mathsf{\left(}\mathsf{\text{o}}\mathsf{\text{n}}\mathsf{\text{e}}\mathsf{\text{s}}\mathsf{\left(}\mathsf{5}\mathsf{\right)}\mathsf{\right)}& \begin{array}{cc}\text{and}& \text{x}=\left[1:5\right]\prime \end{array}\end{array}$$

   The columns of W can be used to form an ordered basis:

   $$F=\left\{{\mathbf{w}}_1,{\mathbf{w}}_2,{\mathbf{w}}_3,{\mathbf{w}}_4,{\mathbf{w}}_5\right\}$$

   Let $L:{ℝ}^5\to {ℝ}^5$ be a linear operator such that

   $$\begin{array}{ccc}L\left({\text{w}}_1\right)={\text{w}}_2,& L\left({\text{w}}_2\right)={\text{w}}_3,& L\left({\text{w}}_3\right)={\text{w}}_4\end{array}$$

   and

   $$\begin{array}{cc}\hfill L\left({\text{w}}_4\right)& =4{\text{w}}_1+3{\text{w}}_2+2{\text{w}}_3+{\text{w}}_4\hfill \\ \hfill L\left({\text{w}}_5\right)& ={\text{w}}_1+{\text{w}}_2+{\text{w}}_3+3{\text{w}}_4+{\text{w}}_5\hfill \end{array}$$

   1. Determine the matrix A representing L with respect to F, and enter it in MATLAB.

   2. Use MATLAB to compute the coordinate vector $\mathbf{y}={\text{W}}^{-1}\mathbf{x}$ of x with respect to F.

   3. Use A to compute the coordinate vector z of L (x) with respect to F.

   4. W is the transition matrix from F to the standard basis for ${ℝ}^5$. Use W to compute the coordinate vector of L (x) with respect to the standard basis.

2. Set $\text{A}=\mathsf{\text{t}}\mathsf{\text{r}}\mathsf{\text{i}}\mathsf{\text{u}}\mathsf{\left(}\mathsf{\text{o}}\mathsf{\text{n}}\mathsf{\text{e}}\mathsf{\text{s}}\mathsf{\left(}\mathsf{5}\mathsf{\right)}\mathsf{\right)}\mathsf{*}\mathsf{\text{t}}\mathsf{\text{r}}$