Vector Space

Traditionally, vector space is regarded as a heterogenous algebra, one have two sets:

• scalars (the field F)
• vectors (the actual space)

with two binary opetations, scalar multiplication and vector addition.

However, it can also be regarded as a homogeneours algebra with an abelian group V equiped with one unary operation \(a: V \to V\) for each element \(a\) of the field F such that

• each \(a\) is a group endomorphism on \(V\), so \((a + b) v = a v + bv\)
• multiplication in F is interpreted as composition, so \((ab)v = a (bv)\)