Theorems · Theorem · linear algebra

rank_matrix_module

∀ (R : Type u) (M : Type v) [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [StrongRankCondition R]
  [Module.Free R M] (m : Type w) (n : Type w') [Finite m] [Finite n],
  Module.rank R (Matrix m n M) =
    Cardinal.lift.{max v w', w} (Cardinal.mk m) * Cardinal.lift.{max v w, w'} (Cardinal.mk n) *
      Cardinal.lift.{max w w', v} (Module.rank R M)

If m and n are finite, the rank of m × n matrices over a module M is (#m).lift * (#n).lift * rank R M.

Defined in

Mathlib.LinearAlgebra.Dimension.Constructions

Cited by

3 results in Mathlib

Foundations

Depth 111 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleStrongRankConditionModule.FreeFiniteFinite