Find a matrix R α ∈ R 2 × 2 such that f ( x ) = f R α ( x ) for every x ∈ R 2

Question

I'm trying to solve this task.

Let α∈R be given. Consider the linear map f:R^2→R^2,

   (x1   -->  ( x1·cos(α)−x2·sin(α)
    x2)        x1.sin(α)+x2·cos(α)).

a) Find a matrix Rα∈R2×2 such that f(x) =f Rα(x) for everyx∈R^2.

b) Interpret the map f, i.e. state what it does to a given vector x∈R^2.

First I thought, I should use this rule |A| = ad − bc , but I think, It's already used . should i gave α a number like zero and then I will have a matrix and that's it?

Answer 1

Hint: The columns of the matrix are $f(1,0)$ and $f(0,1)$.

For the second part, look at what happens to $(1,0)$ and $(0,1)$ and apply a little trigonometry.