Linear Algebra, 5th Edition

1.7* Maximal Linearly Independent Subsets

In this section, several significant results from Section 1.6 are extended to infinite-dimensional vector spaces. Our principal goal here is to prove that every vector space has a basis. This result is important in the study of infinite-dimensional vector spaces because it is often difficult to construct an explicit basis for such a space. Consider, for example, the vector space of real numbers over the field of rational numbers. There is no obvious way to construct a basis for this space, and yet it follows from the results of this section that such a basis does exist.

The difficulty that arises in extending the theorems of the preceding section to infinite-dimensional vector spaces is that the principle ...