# 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^{[1]}
by Saunders Mac Lane
of Steinitz's lemma to matroids.^{[2]}

The Steinitz exchange lemma is a basic result in computational mathematics, especially in linear algebra and in combinatorial algorithms.^{[3]}