SOLUTION: I have a homework question where it asks the following question: Find the standard matrix representation for the following linear transformation. L is the linear transformation th

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Question 20562: I have a homework question where it asks the following question:
Find the standard matrix representation for the following linear transformation. L is the linear transformation that rotates each xER^2 (every x in R^2) by 45 degrees in the counterclockwise direction then reflects about the line x2=x1.
So far, I drew the geometric representation of the problem, where e1 is rotated from x1-axis and e2 is rotated from x2-axis by 45 degrees. Where I am is the 'relfects about the line x2=x1.' I have no idea what to do from here, can you help me?
Answer by khwang(438) About Me  (Show Source):
You can put this solution on YOUR website!
L is the linear transformation that rotates each xER^2 (every x in R^2) by 45 degrees in the counterclockwise direction then reflects about the line x2=x1.

I prefer to use x,y instead of x1, x2.
(x,y)^T means the column vector in R%5E2
the rotation (45 deg) as
(x, y)^T-->
(cospi%2F4 -sinpi%2F4) (x)
(sin pi%2F4 cos pi%2F4) (y)
--->
(x cos pi%2F4 -y sin pi%2F4)
(x sin pi%2F4 + y cos pi%2F4)

-->
(sqrt%282%29%2Ax%2F2 -sqrt%282%29%2Ay%2F2 )
(sqrt%282%29%2Ax%2F2 %2Bsqrt%282%29%2Ay%2F2 )
Then the reflection (x,y)^T-->(y,x)^T (swap x & y)

Hence, L:(x,y)^T=(sqrt%282%29%2Ax%2F2 %2Bsqrt%282%29%2Ay%2F2,sqrt%282%29%2Ax%2F2 -sqrt%282%29%2Ay%2F2 )^T
Kenny