## Linear dependence and linear independence

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 .