3.1 Algebras over Vector Spaces

In this section, we will introduce the concept of algebras over vector spaces.

Table of Contents

• Table of Contents
• Introduction

Introduction

In many physical applications, there are often natural ways to "multiply" vectors together. We can multiply matrices together, we can take the cross product of vectors in , and we can multiply complex numbers together.

Definition 3.1.1 (algebra over a field) An algebra over a field is a vector space over equipped with a bilinear map (called multiplication) . The image of under this map is denoted by for all . This multiplication must satisfy the following properties for all and all :

1. Linearity in the first argument: ,
2. Linearity in the second argument: .