Linear Algebra: Solving Rotation by Angle θ

[theshonen8899]

This is for my Linear Algebra class:

Homework Statement

For an angle θ ∈ [0, 2π), find the linear map fθ : R2 → R2, which describes the rotation
by the angle θ in the counterclockwise direction.

Hint : For a given angle θ, find a, b, c, d ∈ R such that fθ(x1, x2) = (ax1+bx2, cx1+dx2).

Homework Equations

e^(x+yi) = (e^x)*(cos(y) + sin(y)i)

The Attempt at a Solution

Circle in counterclockwise direction is (cosθ, sinθ)
therefore
cosθ = ax1 + bx2
sinθ = cx1 + dx2
i(sinθ = cx1 + dx2) = isinθ = icx1 + idx2

cosθ + isinθ = ax1 + bx2 + icx1 + idx2 = e^(0 + iθ) = e^(iθ)

This isn't much but I've really been working on this problem the entire day and I really have no clue what I'm supposed to do. I feel like a damned fool for having to resort to this but I'd really like to have a solution to this before I head off to my quiz.

Thanks guys.

 

[vela]

Hint: Calculate fθ(1,0) and fθ(0,1).