Math 2.0

Linear and multilinear algebra; matrix theory Vector spaces Lemma

Steinitz exchange lemma

Builds on Linear independenceSpanning set

Statement

Let FF be a field and let VV be a vector space over FF. Let u1,…,um∈Vu_1, \dots, u_m \in V be linearly independent, and let w1,…,wn∈Vw_1, \dots, w_n \in V span VV. Then m≤nm \le n. Moreover, after a renumbering of the wjw_j, the family u1,…,um,wm+1,…,wnu_1, \dots, u_m, w_{m+1}, \dots, w_n spans VV.

Remarks

The lemma follows by induction on mm: each uiu_i replaces one of the wjw_j in a spanning family. A consequence is that any two finite bases of VV have the same number of elements. The lemma is named after Ernst Steinitz.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionSteinitz exchange lemma, statement, no evidence levelSteinitz exchange lem…statement
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  1. Vector space definition
  2. Linear combination definition
  3. Linear subspace definition
  4. Linear independence definition
  5. Linear span definition
  6. Spanning set definition
  7. Steinitz exchange lemma statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Steinitz_exchange_lemma, Statement
About this page
Tag08ZDRN6H41
URIm2://08ZDRN6H41
Aliasesreplacement-theorem
Kindstatement, lemma
WritingW1 stated
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 08ZDRN6H41.
UsesLinear independence, Spanning set
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