Questions: Let T: ℝ² → ℝ² be the linear transformation that first rotates points clockwise through 45° ( π / 4 radians) and then reflects points through the line y=x. Find the standard matrix A for T. A=[ square square square square ]

Solution Steps

To find the standard matrix for the linear transformation \( T \), we need to determine the matrices for each individual transformation and then multiply them. First, find the matrix for the clockwise rotation by \( 45^\circ \). Then, find the matrix for reflection through the line \( y = x \). Finally, multiply these matrices in the correct order to get the standard matrix \( A \).

To rotate a point clockwise by \( 45^\circ \) (or \(-\frac{\pi}{4}\) radians), the rotation matrix is given by: \[ R = \begin{bmatrix} \cos(-\frac{\pi}{4}) & \sin(-\frac{\pi}{4}) \\ -\sin(-\frac{\pi}{4}) & \cos(-\frac{\pi}{4}) \end{bmatrix} \] Substituting the values, we have: \[ R = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \]

The reflection matrix through the line \( y = x \) is: \[ M = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

The standard matrix \( A \) for the transformation \( T \) is obtained by multiplying the reflection matrix \( M \) by the rotation matrix \( R \): \[ A = M \cdot R = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \] Carrying out the matrix multiplication, we get: \[ A = \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \end{bmatrix} \]

Final Answer