Dimension theorem for vector spaces
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In mathematics, the dimension theorem for vector spaces states that a vector space has a definite, well-defined number of dimensions. This may be finite, or an infinite cardinal number.
Formally, the dimension theorem for vector spaces states that
- Given a vector space V, any two linearly independent generating sets (in other words, any two bases) have the same cardinality.
If V is finitely generated, the result says that any two bases have the same number of elements.
The cardinality of a basis is called the dimension of the vector space.
While the proof of the existence of a basis for any vector space in the general case requires Zorn's lemma and is therefore equivalent to the axiom of choice, the uniqueness of the cardinality of the basis requires only the ultrafilter lemma, which is strictly weaker. The theorem can be generalized to arbitrary R-modules for rings R having invariant basis number.
For the finitely generated case it can be done with elementary arguments of linear algebra, requiring no forms of choice.
[edit] Kernel extension theorem for vector spaces
This application of the dimension theorem is sometimes itself called the dimension theorem. Let
- T: U → V
be a linear transformation. Then
- dim(range(T)) + dim(kernel(T)) = dim(U),
that is, the dimension of U is equal to the dimension of the transformation's range plus the dimension of the kernel. See rank-nullity theorem for a fuller discussion.
