nLab
vector space

Context

Higher linear algebra

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

A for kk a field, a vector space over kk is module over the ring kk. Sometimes a vector space over kk is called a kk-linear space. (Compare ‘kk-linear map’.)

The category of vector spaces is typically denoted Vect, or Vect kVect_k if we wish to make the field kk explicit. So

Vect kkMod. Vect_k \coloneqq k Mod \,.

This category has vector spaces over kk as objects, and kk-linear maps between these as morphisms.

Properties

By the basis theorem (and using the axiom of choice) every vector space admits a basis.

Revised on October 6, 2013 21:22:50 by Urs Schreiber (195.37.209.182)