Steinitz exchange lemma

The Steinitz exchange lemma states that a linearly independent set in a vector space cannot be larger than a finite spanning set. It also shows how to replace spanning-set vectors with independent-set vectors while preserving spanning.

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Linear independence: A set of vectors is linearly independent when only the trivial linear combination equals zero. The lemma begins with vectors whose independence forces each exchange to use a distinct spanning vector.
Vector space: A vector space is a set with addition and scalar multiplication satisfying axioms that support linear combinations. The lemma applies to vectors and linear combinations within this algebraic structure.
Ernst Steinitz: Ernst Steinitz was a German mathematician whose work helped establish the abstract theory of fields. The exchange lemma bears his name and is associated with his foundational algebraic work.
Matrix rank: The rank of a matrix is the dimension of its column space, equivalently its row space. Exchange explains why independent columns cannot outnumber columns in a spanning representation.
Schroeder–Bernstein theorem: The Schroeder–Bernstein theorem states that injections in both directions between two sets imply a bijection. It compares cardinalities by injections, whereas Steinitz exchange uses linear combinations and finite spanning.
Spanning set: A set of vectors spans a vector space when every vector in the space is a linear combination of its members. The lemma replaces members of a finite spanning set without losing its ability to span.
Linear combination: A linear combination is a finite sum of vectors multiplied by scalars. The exchange step expresses a chosen independent vector using the current spanning set.
Invariant basis number: A ring has invariant basis number if free modules of different finite ranks are never isomorphic. The vector-space exchange argument is a model for proving rank invariance in broader algebraic settings.
Column space: The column space of a matrix is the span of its column vectors. Applying exchange to the columns bounds the number of independent columns by the total number supplied.
Infinite-dimensional vector space: An infinite-dimensional vector space has no finite spanning set. The lemma's finite cardinality bound does not directly compare arbitrary infinite independent and spanning sets.