Steinitz exchange lemma


The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements. The result is named after the German mathematician Ernst Steinitz. The result is often called the Steinitz–Mac Lane exchange lemma, also recognizing the generalization
by Saunders Mac Lane
of Steinitz's lemma to matroids.

Statement

If is a set of linearly independent vectors in a vector space, and span, then:
1. ;
2. The set spans, possibly after reordering the.

Proof

Suppose that we have the indicated sets of vectors. We wish to show that for each, we have that, and that the set spans . We proceed by mathematical induction on.
For the base case, suppose is zero.
In this case, the claim holds because there are no vectors, and the set spans by hypothesis.
For the inductive step, assume the proposition is true for some. Since, and spans , there exist coefficients such that
At least one of must be non-zero, since otherwise this equality would contradict the linear independence of ; note that this additionally implies that. By reordering the, we may assume that is not zero. Therefore, we have
In other words, is in the span of. The latter span therefore contains each of the vectors, and hence must contain the span of these latter vectors as a subset. But since the latter span is , this simply means that the span of contains as a subset. We have therefore shown that our claim is true of, completing the inductive step.
We have thus shown that for each, we have that, and that the set spans .
The fact that follows from setting in this result.

Applications

The Steinitz exchange lemma is a basic result in computational mathematics, especially in linear algebra and in combinatorial algorithms.