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Sets larger than a finitely generated space's bases are linearly dependent

Sets larger than a finitely generated space's bases are linearly dependent

Jul 27, 20231 min read

  • mathematics

up:: Steinitz Exchange Lemma

Let (V,+,⋅) be a Finitely Generated Vector Space, with Spanning Set S={s1​,…,sn​}. Let U={u1​,…,um​}⊆V.

From the Steinitz Exchange Lemma, by contrapositive, we have that m>n implies that U is Linearly Dependent (assuming S is a spanning set as hypothesis).


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Backlinks

  • 021a MOC Linear Algebra
  • Every linearly independent set of a finitely generated vector space has at most the same number of vectors as its spanning set
  • Hamel Basis

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