fin_two_smul_prod — Mathlib

English

Let g ∈ GL2(R) and v = (v1,v2) ∈ R×R. The action is again the usual matrix action on vectors: g • v = (g11 v1 + g12 v2, g21 v1 + g22 v2).

Русский

Пусть g ∈ GL2(R) и v ∈ R×R. Действие то же самое: g • v = (g11 v1 + g12 v2, g21 v1 + g22 v2).

LaTeX

$$$g\\cdot (v_1,v_2) = (g_{00}v_1+g_{01}v_2,\\; g_{10}v_1+g_{11}v_2).$$$

Lean4

@[simp]
theorem fin_two_smul_prod {R : Type*} [CommRing R] (g : GL (Fin 2) R) (v : R × R) :
    g • v = (g 0 0 * v.1 + g 0 1 * v.2, g 1 0 * v.1 + g 1 1 * v.2) := by simp [Units.smul_def]

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