cRank_toNat_eq_finrank — Mathlib

English

The natural value of the cardinal cRank is equal to the finite dimension of the column space, i.e., A.cRank.toNat = finrank_R(span_R(range A.col)).

Русский

Естественная величина кардинала cRank равна размерности пространства столбцов: A.cRank.toNat = finrank_R(span_R(range A.col)).

LaTeX

$$$A.cRank.toNat = \operatorname{finrank}_R\big(\operatorname{span}_R(\operatorname{range}(A^{\top}))\big).$$$

Lean4

theorem cRank_toNat_eq_finrank (A : Matrix m n R) : A.cRank.toNat = Module.finrank R (span R (range A.col)) :=
  rfl

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