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 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Steinitz exchange lemma, Wikipedia
Sources
1. Steinitz exchange lemma, Wikipedia
Lead, first sentence
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.
In Branch: Linear Algebra, Lead sentence
The Steinitz exchange lemma is a theorem in linear algebra concerning bases, dimensionality of a vector space, stating that for an
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.