Most applications of mathematics are involved with the concept of measurement and hence of the magnitude or relative size of various quantities. So it is not surprising that the fields of real and complex numbers, which have a built-in notion of distance, should play a special role. Except for Section 6.8, in this chapter we assume that all vector spaces are over either the field of real numbers or the field of complex numbers. See Appendix D for properties of complex numbers.

We introduce the idea of distance or length into vector spaces via a much richer structure, the so-called inner product space structure. This added structure provides applications to geometry (Section 6.5 and 6.11), physics (Section 6.9), conditioning in systems of linear equations (Section 6.10), least squares (Section 6.3), and quadratic forms (Section 6.8).