Linear Algebra - Matrix Transformations

[01KD]

Homework Statement

Let L denote the line through the origin in R2 that that makes angle -∏ < theta ≤ ∏
with the positive x-axis. The reflection operator that reflects points about L in R2
is the matrix transformation R2 --> R2 with standard matrix

[cos 2(theta) sin 2(theta); sin 2(theta) -cos 2(theta)]

Show that the composition of a rotation operator followed by a re
reflection operator is another reflection operator.

Homework Equations

Standard matrix for reflection in R2 comes to mind;

[cos(theta) -sin(theta); sin (theta) cos (theta)]

These are 2x2 matrices and I'm not sure how to input them here so I put ; to separate the two columns.

The Attempt at a Solution

I'm kind of confused as to how to attempt to solve the question. I mean I could use the 2 standard matrices and use matrix multiplication which gives a really ugly 2x2 that I'm not sure what to do with. Any help to get me started would be appreciated :).

 

[01KD]

Anybody? : (

 

[HallsofIvy]

It's pretty straight forward isn't it? You know how to write the rotation and reflection as matrices. The "rotation followed by a reflection" is the product of those two matrices. Do that multiplication and show that it is a reflection by determining the appropriate angle for the "line of reflection"?

 

[01KD]

Thanks HallsofIvy! It turns out that I read the question wrong. I just had to use matrix multiplication and simplify. When I did it yesterday, I wasn't sure about how to simplify it. But using trig identities makes it very simple. Thanks again for your help!