Question

Find the matrix A of the linear transformation T from R2 to R2 that rotates any vector through an angle of 45∘ in the counterclockwise direction. A=[]

Answer 1

The matrix A of the linear transformation T that rotates vectors through an angle of 45 degrees in the counterclockwise direction is [sqrt(2)/2 -sqrt(2)/2] [sqrt(2)/2 sqrt(2)/2].

Step-by-step explanation:

To find the matrix A of the linear transformation T that rotates vectors through an angle of 45 degrees in the counterclockwise direction, we can use the rotation matrix formula. The rotation matrix for a 2D transformation is:

R = [cos(theta) -sin(theta)]

[sin(theta) cos(theta)]

Substituting theta = 45 degrees into the formula, we get:

A = [cos(45) -sin(45)]

[sin(45) cos(45)]

Simplifying, this becomes:

A = [sqrt(2)/2 -sqrt(2)/2]

[sqrt(2)/2 sqrt(2)/2]