Let T R2 rightarrow R2 be the linear transformation that fir

Let T: R^2 rightarrow R^2 be the linear transformation that first reflects across the line y = 2x and then rotates pi/4 counterclockwise. Find the standard matrix of T.

Solution

We know that matrix for reflection across the line y = mx is 1/(1+m²)A where A =

1-m²

2m

2m

m² -1

When m = 2, the required matrix is M =

-3/5

4/5

4/5

3/5

Further, the matrix for a counterclockwise rotation by /4 is N=

cos45⁰

sin45⁰

-sin45⁰

cos45⁰

or N=

1/2

1/2

-1/2

1/2

Then the required matrix is NM =

-3/52+22/5

3/52+22/5

3/52+22/5

3/52 -22/5

1-m2	2m
2m	m2 -1