## Linear dependence and linear independence

Posted December 1st, 2007 by Isoscel

Let be a -vector space over a field .

A set is called system of generators for V if and only if

exists , such as

A set is called linear independent if and only if

from

we have

Definition

Basis of vector space.

A set is called basis if it is linear independent and system of generators.

Theorem

Let be a -vector space over a field , and two basis of V. There exists a bijective function .

Definition

We call dimension of a -vector space the cardinal of a basis.

A finite dimensional space is a space which has a finite basis.

Theorem.

Any finite n-dimensional vector space is isomorph to .