Viewing an algebraic number field as a vector space relative to a subfield, which was foreshadowed in Chapter 4, involves varying the field of "scalars" in the definition of vector space. This leads in turn to relative concepts of "basis" and "dimension" which must be taken into account in algebraic number theory. In this chapter we review linear algebra from the ground up, with an emphasis on the relative point of view. This brings some nonstandard results into the picture, such as the Dedekind product theorem and the representation of algebraic numbers by matrices.