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Linear dependence and linear independence

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


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


we have

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

Basis of vector space.
A set $ (e_i)_{i\in I} $ is called basis if it is linear independent and system of generators.
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 $.
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.
Any finite n-dimensional vector space is isomorph to $ \mathcal{K}^n $.

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