(Linear Algebra) How to determine the expression of this linear mapping ?

Wang Feng: (Linear Algebra) How to determine the expression of this linear mapping ?
Could someone please show me the solution to this problem in Linear algebra? Thanks a lot!

Determine the expression of the linear mapping in the plane, which reflects all points with respect to the straight line y=2x.

Answers and Views:

Answer by cheeser1
If you can find a basis for R2 and figure out what this does to the basis, you are done.

There are two approaches:

      • Use the standard basis — calculations are harder, but the matrix is easy to find
      • Use a “nice” basis — calculations are easy, but the matrix is harder to find.

I will do the first method. Where does (1,0) go when reflected by that line? It goes to (-3/5,4/5).

And where does (0,1) go? It goes to (4/5,3/5).

So the matrix is simply:

[ -3/5 4/5 ]
[ 4/5 3/5 ]

which are those two things, put into the columns.