Pick any [0, ) and define a linear transformation T : R 2 R

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.

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To show that T is angle preserving we must show that for all x y and a b in R2 the angle between x y ... View the full answer