Find the standard matrix of the linear transformation T: R_2 --->R_2 if T first rotates points pi/4 radians

linear-tranformation
standard-matrix

#1

Find the standard matrix of the linear transformation T: R_2 —>R_2 if T first rotates points pi/4 radians (counter clockwise) and then reflects points through the horizontal x1 axis.

Answer:

The transformation matrix, we will call this T, is the combination of two transformations. First, we must find the matrix of the rotation transformation (A), and second, the matrix of the reflection (B). Once we find both of those, we take the product (AB) to get to our matrix T. The matrix of the rotation will be 2x2 with the first column being [cos(pi/4),sin(pi/4)] and the second column being [-sin(pi/4),cos(pi/4)]. The matrix of the reflection will be 2x2 with the first column being [1,0] and the second being [0,-1]. When we take the product of these two transformations, we end up with our final matrix T which has column 1 being [(sqrt(2)/2),(sqrt(2)/2)] and column 2 being [(sqrt(2)/2),(-sqrt(2)/2)]