1 - Preliminaries

Summary

1.1 Review of Linear Algebra

Linear algebra is primarily the study of finite dimensional vector spaces and linear transformations between them. In a first course, one encounters the fact that every such vector space has a basis, that linear transformations may be associated to matrices, and one finally understands why matrix multiplication looks so clumsy. This section sets out to relive those glory days.

1.1.1 DEFINITION A vector space over a field K is a set E together with two operations a: E × E → E (vector addition) and s: K × E → E (scalar multiplication) written as a(x, y) = x + y and s(α, x) = αx, which satisfy the following properties.