I have a linear map R α = ( c o s α − s...

Raegan Bray

Raegan Bray

Answered

2022-07-16

I have a linear map R α = ( c o s α s i n α s i n α c o s α ) and the question goes like, how does R α transform a vector?
Can someone please help me?

Answer & Explanation

nezivande0u

nezivande0u

Expert

2022-07-17Added 16 answers

The comments above on the OP are good, but here's another perspective. How does R α change the length of a vector? It is easy to verify that
R α T R α = ( cos 2 α + sin 2 α 0 0 cos 2 α + sin 2 α ) = I
So that for some vector v we have | | R α v | | 2 = ( R α v ) T ( R α v ) = v T R α T R α v = v T I v = v T v = | | v | | 2 . That is, R α is an isometry (it can't change the length of vectors). So if R α cannot change the length of a vector, it could only possibly rotate a vector. So what is this angle? We can compute the angle, θ, between R α v and v by computing
cos θ = ( R α v ) v | | R α v | | | | v | | .
I'll leave it to you to verify that this gives θ = α (mod π or whatever)

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