The Steinitz Exchange Lemma states that if a set of vectors spans a vector space and a second, linearly independent set of vectors lies in that space, then each vector of the independent set can be exchanged for a vector of the spanning set while keeping the result a spanning set, so in particular the independent set can never be larger than the spanning one. Named for Ernst Steinitz, it is the basic combinatorial fact underlying the proof that every vector space has a well-defined dimension.
Facts
StatementFor any set smaller than a spanning set, there is a set of vectors in the spanning set but missing from the smaller set that can be added to the smaller set to make it spanning as well. 1 Classification
Statement Form Sources
1. Steinitz exchange lemma, Wikipedia
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The Steinitz exchange lemma is a theorem in linear algebra concerning bases, dimensionality of a vector space, stating that for any set smaller than a spanning set, there is a set of vectors in the spanning set but missing from the smaller set that can be added to the smaller set to make that set spanning as well.
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