Serlo: EN: Steinitz's theorem

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We prove the exchange theorem to show well-definedness of the term "dimension" later.

Motivation

In this article we will discuss the exchange lemma and Steinitz's exchange theorem. These state how a given basis of a vector space can be converted into another one by cleverly replacing some of the old basis vectors with new vector space elements. This is especially useful when you want to construct a basis that contains certain previously fixed vectors. Another consequence of the replacement theorem is the fact that linearly independent sets have a lower or equal cardinality than bases. This result is essential for the definition of the dimension of a vector space. We first prove the exchange lemma and then Steinitz's theorem.

Exchange lemma

The exchange lemma

Next, we prove a slight modification of the exchange lemma. It shows that the lemma is "almost always" applicable. We only assume that the new basis vector $w$ is not the zero vector:

Steinitz's exchange theorem

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