Theorems · Definition · linear algebra

Module.Basis.extendLe

{K : Type u_3} →
  {V : Type u_4} →
    [inst : DivisionRing K] →
      [inst_1 : AddCommGroup V] →
        [inst_2 : Module K V] →
          {s t : Set V} →
            (hs : LinearIndepOn K id s) → (hst : s ⊆ t) → ⊤ ≤ Submodule.span K t → Module.Basis (↑(hs.extend hst)) K V

If s is a family of linearly independent vectors contained in a set t spanning V, then one can get a basis of V containing s and contained in t.

Defined in

Mathlib.LinearAlgebra.Basis.VectorSpace

Cited by

5 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModule