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Algebras

Abstract

The definition an algebra.

Keywords:VectorsVector SpacesAlgebra.

Algebra Definition

Let F\mathbb{F} be a field over which a vector space VV has been defined. Let ψ,σ,ω∈V\psi, \sigma, \omega \in V and a,b∈F1a,b \in \mathbb{F}^1.

The vector space VV is an algebra over F\mathbb{F} if there exists a binary operation,

V×V→V,V \times V \rightarrow V,
(1)

meaning it takes two elements of VV as input and maps them to a single element of VV, that satisfies the following conditions

  1. (ψ+σ)×ω=ψ×ω+σ×ω(\psi + \sigma) \times \omega = \psi \times \omega + \sigma \times \omega

    • Right distributivity

  2. ω×(ψ+σ)=ω×ψ+ω×σ\omega \times (\psi + \sigma) = \omega \times \psi + \omega \times \sigma

    • Left distributivity

  3. (aψ)×(bσ)=(ab)(ψ×σ)(a \psi) \times (b \sigma) = (ab) (\psi \times \sigma)

    • Compatibility with scalars

The above three conditions can be compactly summarised as saying the binary operator in the definition of the algebra, ×\times, must be bilinear.

Examples

Cross Product
Complex Numbers

The vector space of R3\mathbb{R}^3, defined over the field of real numbers R\mathbb{R} with bilinear product the cross product.

This example can be used to understand the difference between a algebra and an inner product space:

  • R3\mathbb{R}^3 with a cross product →\rightarrow algebra

  • R3\mathbb{R}^3 with a dot product →\rightarrow inner product space

Linear Algebra
Norms
Linear Algebra
Glossary