Dimension theorem for vector spaces
|
|
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.
The proof in the general case makes use of Zorn's lemma (or equivalently, the axiom of choice), while for the finitely generated case it can be done with elementary arguments of linear algebra.
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.