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A vector space on the field $\mathbb{K}$ is a nonempty set V with two operations. $$ + \; : V \times V \rightarrow V$$ $$(u,v) \longrightarrow u+v$$ and $$ \cdot \; : \mathbb{K} \times v \rightarrow V $$ $$(\alpha,v) \longmapsto \alpha\cdot v$$ such that for all $u,v,w \in V$ and $ \alpha, \beta \in \mathbb{K} $ the following properties holds:

  1. $u+v = v+ u$. (Commutative)
  2. $u+ (v+w)= (u+v)+w.$ (Asociative)
  3. There is an only element $0_{V} \in V$, called null vector in $V$, such that $u+0_V= u$ for all $ u \in V$. (Null Element)
  4. For every $ u \in V$, there is an only $u' \in V$, called symetric vector de $u$, such that $u + u' = 0_V.$ (Symetric Element)
  5. $\alpha (u +v ) = \alpha u + \alpha v.$ (Distributivity, element of field)
  6. $(\alpha + \beta) u = \alpha u+ \beta u.$ (Distributivity, element of space)
  7. $\alpha (\beta u)= (\alpha \beta) u = \alpha \beta u.$
  8. $1u= u.$

The elements in $V$ will be called vectors and the elements in $\mathbb{K}$ scalars.