Theorems · Theorem · linear algebra

linearIndependent_iff_card_eq_finrank_span

∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [OrzechProperty R] [inst_2 : AddCommMonoid M] [inst_3 : Module R M]
  [Nontrivial R] {ι : Type u_3} [inst_5 : Fintype ι] {b : ι → M},
  LinearIndependent R b ↔ Fintype.card ι = Set.finrank R (Set.range b)

A finite family of vectors is linearly independent if and only if its cardinality equals the dimension of its span.

Defined in

Mathlib.LinearAlgebra.Dimension.OrzechProperty

Cited by

5 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringOrzechPropertyAddCommMonoidModuleNontrivialFintype