Theorems · Theorem · linear algebra

Submodule.range_ker_disjoint

∀ {ι : Type u'} {R : Type u_2} {M : Type u_4} {M' : Type u_5} {v : ι → M} [inst : Semiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid M'] [inst_3 : Module R M] [inst_4 : Module R M'] {f : M →ₗ[R] M'},
  LinearIndependent R (⇑f ∘ v) → Disjoint (Submodule.span R (Set.range v)) f.ker

If v is an injective family of vectors such that f ∘ v is linearly independent, then v spans a submodule disjoint from the kernel of f. TODO : LinearIndepOn version.

Defined in

Mathlib.LinearAlgebra.LinearIndependent.Basic

Cited by

2 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModule