nLab vector space

Theorems

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Contents

Definition

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

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

$Vect_k \coloneqq k Mod \,.$

This category has vector spaces over $k$ as objects, and $k$-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)