Question 20562
 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^2}}}
  the rotation (45 deg) as
  (x, y)^T-->
(cos{{{pi/4}}}  -sin{{{pi/4}}}) (x)  
(sin {{{pi/4}}}  cos {{{pi/4}}}) (y) 
 --->
  (x cos {{{pi/4}}} -y sin {{{pi/4}}})  
  (x sin {{{pi/4}}} + y cos {{{pi/4}}})
    
 -->
 ({{{sqrt(2)*x/2}}}  {{{-sqrt(2)*y/2}}} )
 ({{{sqrt(2)*x/2}}}  {{{+sqrt(2)*y/2}}} )

 Then the reflection (x,y)^T-->(y,x)^T (swap x & y)
  
 Hence, L:(x,y)^T=({{{sqrt(2)*x/2}}} {{{+sqrt(2)*y/2}}},{{{sqrt(2)*x/2}}} {{{-sqrt(2)*y/2}}} )^T

 Kenny