Theorem. If a vector space has a basis of vectors, then any collection of vectors, where , is going to be linearly dependent.
(We get this theorem "for free" because of the isomorphism between and that we previously showed exists.)
Theorem. If a vector space has a basis of vectors, then any other basis of that vector space will also have vectors.
Proof. Let be a vector space, and let be a basis for with vectors. Let be another basis for .
Thus has exactly elements.
This theorem indicates that the number of elements in a basis is invariant for a vector space – it’s the same for every basis of that vector space.
Definition. If is spanned by a finite set, we say that is finite dimensional, and the dimension of , or , is the number of vectors in a basis of . The dimension of is defined to be .
If is spanned by an infinite set, we say that is infinite dimensional.
Ex. Let be the space of matrices over (that is, the matrices whose entries are real numbers). is a vector space over the real numbers. What is ?
If we can find a basis for , then all we have to do is count the number of vectors in that basis to determine .
spans and is linearly independent, so is a basis for .
contains four elements, thus .
Theorem. Let be a subspace of a finite dimensional vector space . Any linearly independent set in can be expanded if needed to be a basis of . (You can keep adding linearly independent vectors to the set to eventually get a vector of .) Also, is finite dimensional and .