Theorems · Definition · linear algebra

setBasisOfLinearIndependentOfCardEqFinrank

{K : Type u} →
  {V : Type v} →
    [inst : DivisionRing K] →
      [inst_1 : AddCommGroup V] →
        [inst_2 : Module K V] →
          {s : Set V} →
            [Nonempty ↑s] →
              [inst_4 : Fintype ↑s] →
                LinearIndependent K Subtype.val → s.toFinset.card = Module.finrank K V → Module.Basis (↑s) K V

A linear independent set of finrank K V-many vectors forms a basis.

Defined in

Mathlib.LinearAlgebra.FiniteDimensional.Lemmas

Cited by

2 results in Mathlib

Foundations

Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleNonemptyFintype