Theorems · Theorem · linear algebra

exists_linearIndependent_cons_of_lt_rank

∀ {R : Type u_1} {M : Type u} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M]
  [HasRankNullity.{u, u_1} R] [StrongRankCondition R] {n : ℕ} {v : Fin n → M},
  LinearIndependent R v → ↑n < Module.rank R M → ∃ x, LinearIndependent R (Fin.cons x v)

Given a family of n linearly independent vectors in a space of dimension > n, one may extend the family by another vector while retaining linear independence.

Defined in

Mathlib.LinearAlgebra.Dimension.RankNullity

Cited by

2 results in Mathlib

Foundations

Depth 116 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleHasRankNullityStrongRankCondition