Theorems · Theorem · linear algebra
linearIndependent_iff_finset_linearIndependent
∀ {ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [inst : Semiring R] [inst_1 : AddCommMonoid M]
[inst_2 : Module R M], LinearIndependent R v ↔ ∀ (s : Finset ι), LinearIndependent R (v ∘ Subtype.val)A family is linearly independent if and only if all of its finite subfamily is linearly independent.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- Finsetstatement and proof · cited by 13,712
- AddCommMonoidstatement and proof · cited by 12,281
- Finset.sumproof · cited by 5,195
- LinearIndependentstatement and proof · cited by 560
- Subtype.val_injectiveproof · cited by 232
- LinearIndependent.compproof · cited by 41
- Finset.sum_coe_sortproof · cited by 24
- linearIndependent_iff'ₛproof · cited by 9
- Fintype.linearIndependent_iffₛproof · cited by 5
Cited by2
Results whose statement or proof uses this declaration.
- linearIndepOn_iff_linearIndepOn_finsetproof · cited by 1
- LinearIndependent.map_pow_expChar_pow_of_isSeparableproof · cited by 1