Linear dependence and linear independence

Submitted by Isoscel on Sat, 12/01/2007 - 12:56.

Let $(V,+,*)$ be a $\mathcal{K}$ -vector space over a field $\mathcal{K}$ .
A set $(g_i)_{i\in I}$ is called system of generators for V if and only if
$\forall x\in V$ exists $n\in N$ , $i_k\in I,\: \alpha_k\in \mathcal{K},\:1\leq k\leq n$ such as

$x=\sum_{k=1}^{n}\alpha_k*{g_i}_k$

A set $(e_i)_{i\in I}$ is called linear independent if and only if
$\forall n\in N, \forall \alpha_k\in \mathcal{K}, , \forall i_k \in I, \:1\leq k\leq n,\:$ from

$\sum_{k=1}^{n}\alpha_k*{e_i}_k=0_{V}$

we have

$\forall 1\leq k\leq n,\:\alpha_k=0_{\mathcal{K}}$

Definition
Basis of vector space.
A set $(e_i)_{i\in I}$ is called basis if it is linear independent and system of generators.
Theorem
Let $(V,+,*)$ be a $\mathcal{K}$ -vector space over a field $\mathcal{K}$ , $(e_i)_{i\in I}$ and $(e_j)_{j\in J}$ two basis of V. There exists a bijective function $\phi:I\rightarrow J$ .
Definition
We call dimension of a $\mathcal{K}$ -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 $\mathcal{K}^n$ .

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Linear dependence and linear independence