Pick any [0, ) and define a linear transformation T : R 2 R 2 by the matrix T = cos() sin()

Pick any θ ∈ [0, π) and define a linear transformation T : R 2 → R 2 by the matrix T =  cos(θ) sin(θ) − sin(θ) cos(θ) . That is, T(x, y) = (cos(θ)x+ sin(θ)y, − sin(θ)x+ cos(θ)y. Show that T is angle preserving and if (x, y) 6= (0, 0), then the angle between (x, y) and T(x, y) is precisely θ. Such a linear transformation is called a rotation.