l Assume that T R2 R2 is a linear transformation that rotate

l) Assume that T: R2 R2 is a linear transformation that rotates all the points in R through radians (clockwise), then reflects points through the vertical x aXIX. and then reflects by the line x -x1

Solution

Ans-

We only need to consider the action of T on the two standard basis vectors. For e1, the 3/4 clockwise rotation, puts the vector in the third quadrant, /4 radians from the negative x1-axis. Recall from trigonometry that the coordinates for this point on the unit circle are ( 2/2, 2/2). Reflecting this point across the x1-axis changes the sign of the x2-coordinate but doesn’t change the x1-coordinate. So this transformation maps e1 to ( 2/2, 2/2) (notice that this is the same as the coordinate given in the hint). Similarly, e2 is first rotated into the fourth quadrant to the point ( 2/2, 2/2) and then reflected to the point ( 2/2, 2/2). And the standard matrix of T must be A = 2/2 2/2 2/2 2/2