Chapter 7.2, Problem 30P

(a)

To determine

To show: The linear transformation y rotates in the counterclockwise rotation of the Cartesian system.

(b)

To determine

To show: The matrix ${\text{A}}^n=\left[\begin{array}{cc}\cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \end{array}\right]$ is true.

(c)

To determine

To show: The expression $\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]\left[\begin{array}{cc}\cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{array}\right]=\left[\begin{array}{cc}\cos \left(\alpha +\beta \right)& -\sin \left(\alpha +\beta \right)\\ \sin \left(\alpha +\beta \right)& \cos \left(\alpha +\beta \right)\end{array}\right]$ is true.

(d)

To determine

The effect of the scalar matrix on the diagonal matrix.

(e)

To determine

To give: The geometrical meaning of $\text{y}=\text{Ax}$ for the given matrices.