Theorems · Theorem · linear algebra

LinearIndependent.map_injOn

∀ {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'],
  LinearIndependent R v →
    ∀ (f : M →ₗ[R] M'), Set.InjOn ⇑f ↑(Submodule.span R (Set.range v)) → LinearIndependent R (⇑f ∘ v)

If a linear map is injective on the span of a family of linearly independent vectors, then the family stays linearly independent after composing with the linear map. See LinearIndependent.map for the version with Set.InjOn replaced by Disjoint when working over a ring.

Defined in

Mathlib.LinearAlgebra.LinearIndependent.Basic

Cited by

7 results in Mathlib

Foundations

Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidAddCommMonoidModuleModule