Chapter 2 Vector spaces

4cm Mathematics is an art, but there are stricter rules than in other arts.

– John Tate (1925 - )

Abel prize winner

Linear algebra is the study of vector spaces and of the linear maps between them. In this Chapter we will begin with an abstract definition of a vector space, and then introduce some of the most fundamental concepts in linear algebra, including subspaces, linear independence, span, and coordinates.

Vector spaces are defined to mimic something that is well understood, namely, the set of vectors in ${ℝ}^n$. The purpose of making this abstract definition is to make the theory of linear algebra accessible for a wide range of applications, including those not necessarily involving ${ℝ}^n$.