Question

Find the standard matrix A for the linearly transformation T :R^2_R^2

Answer 1

Find the standard matrix of $T:\R^2\to \R^2$ rotates points (about the origin) through $−π/4$ radians (clockwise).

Let $T:\R^2\to \R^2$ be a linear transformation given by rotating vectors in the counter-clockwise direction through an angle of $θ.$

Then the matrix $A$ of $T$ is given by

$$T\bigg(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\bigg)=\begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix}$$

$$T\bigg(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\bigg)=\begin{pmatrix} - \sin \theta \\ \cos \theta \end{pmatrix}$$

$$A=\begin{pmatrix} \cos \theta & - \sin\theta \\ \sin \theta & \cos \theta \end{pmatrix}$$

Given $\theta=\pi/4.$ Then

$$A=\begin{pmatrix} \cos (\pi/4) & - \sin (\pi/4) \\ \sin (\pi/4) & \cos(\pi/4) \end{pmatrix}=\begin{pmatrix} \sqrt{2}/2& -\sqrt{2}/2 \\ \sqrt{2}/2 & \sqrt{2}/2 \end{pmatrix}$$