The Dimension of a Vector Space

In the prior subsection we defined the basis of a vector space, and we saw that a space can have many different bases. For example, following the definition of a basis, we saw three different bases for $\mathbb {R} ^{2}$. So we cannot talk about "the" basis for a vector space. True, some vector spaces have bases that strike us as more natural than others, for instance, $\mathbb {R} ^{2}$'s basis ${\mathcal {E}}_{2}$ or $\mathbb {R} ^{3}$'s basis ${\mathcal {E}}_{3}$ or ${\mathcal {P}}_{2}$'s basis $\langle 1,x,x^{2}\rangle$. But, for example in the space $\{a_{2}x^{2}+a_{1}x+a_{0}\,{\big |}\,2a_{2}-a_{0}=a_{1}\}$, no particular basis leaps out at us as the most natural one. We cannot, in general, associate with a space any single basis that best describes that space.

We can, however, find something about the bases that is uniquely associated with the space. This subsection shows that any two bases for a space have the same number of elements. So, with each space we can associate a number, the number of vectors in any of its bases.

This brings us back to when we considered the two things that could be meant by the term "minimal spanning set". At that point we defined "minimal" as linearly independent, but we noted that another reasonable interpretation of the term is that a spanning set is "minimal" when it has the fewest number of elements of any set with the same span. At the end of this subsection, after we have shown that all bases have the same number of elements, then we will have shown that the two senses of "minimal" are equivalent.

Before we start, we first limit our attention to spaces where at least one basis has only finitely many members.

Definition 2.1

A vector space is finite-dimensional if it has a basis with only finitely many vectors.

(One reason for sticking to finite-dimensional spaces is so that the representation of a vector with respect to a basis is a finitely-tall vector, and so can be easily written.) From now on we study only finite-dimensional vector spaces. We shall take the term "vector space" to mean "finite-dimensional vector space". Other spaces are interesting and important, but they lie outside of our scope.

To prove the main theorem we shall use a technical result.

Again, although we sometimes say "finite-dimensional" as a reminder, in the rest of this book all vector spaces are assumed to be finite-dimensional. An instance of this is that in the next result the word "space" should be taken to mean "finite-dimensional vector space".

The main result of this subsection, that all of the bases in a finite-dimensional vector space have the same number of elements, is the single most important result in this book because, as Example 2.9 shows, it describes what vector spaces and subspaces there can be. We will see more in the next chapter.