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Steinitz exchange lemma
Summary
The Steinitz exchange lemma is a foundational theorem in linear algebra stating that if $U$ is a linearly independent set of vectors and $W$ is a spanning set for a vector space, then $U$ cannot contain more elements than $W$ ($\vert{}U\vert{} \le \vert{}W\vert{}$), and a subset of elements from $W$ can be replaced by elements of $U$ to maintain a complete spanning set.
Named after German mathematician Ernst Steinitz (and extended to matroids as the Steinitz–Mac Lane exchange lemma), it is a crucial tool used to prove that every basis of a finite-dimensional vector space has the exact same number of elements, thereby providing a rigorous foundation for the concept of dimension.
ARTICLE
Summary
The Steinitz exchange lemma is a foundational theorem in linear algebra stating that if $U$ is a linearly independent set of vectors and $W$ is a spanning set for a vector space, then $U$ cannot contain more elements than $W$ ($\vert{}U\vert{} \le \vert{}W\vert{}$), and a subset of elements from $W$ can be replaced by elements of $U$ to maintain a complete spanning set.
Named after German mathematician Ernst Steinitz (and extended to matroids as the Steinitz–Mac Lane exchange lemma), it is a crucial tool used to prove that every basis of a finite-dimensional vector space has the exact same number of elements, thereby providing a rigorous foundation for the concept of dimension.
ARTICLE
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