English
The rotation by angle θ on E×E is the linear map (x, y) ↦ (cos θ x + sin θ y, -sin θ x + cos θ y).
Русский
Поворот на угол θ в пространстве E×E задаётся линейным отображением (x, y) ↦ (cos θ x + sin θ y, -sin θ x + cos θ y).
LaTeX
$$$ \mathrm{rotation}(\theta): E \times E \to L_{\mathbb{R}} E \times E \quad (x, y) \mapsto (\cos\theta \cdot x + \sin\theta \cdot y, -\sin\theta \cdot x + \cos\theta \cdot y). $$$
Lean4
theorem _root_.ContinuousLinearMap.rotation_apply (θ : ℝ) (x : E × E) :
ContinuousLinearMap.rotation θ x = (Real.cos θ • x.1 + Real.sin θ • x.2, -Real.sin θ • x.1 + Real.cos θ • x.2) :=
rfl