Definition : A vector space is a nonempty set V with two operation (addition, scalar, multiplication ) that satisfy the following conditions:
\[\begin{array}{l}
1)u + v\;belongs\;V\;\;\;\;\;\;whenever\;u,v\;belongs\;V\\\\
2)u + v = v + u\;\;\;\;\;\;for\;all\;\;u,v\;belongs\;V\\\\
3)\left( {u + v} \right) + w = u + \left( {v + w} \right)\;\;\;for\;all\;u,v,w\;belongs\;V\\\\
4)\;There\;\;exists\;{0_v}belongs\;V\;such\;that\;{0_v} + u = u\\\\
5)for\;all\;u\;belongs\;V\;,\;there\;exists\;\left( { - u} \right)belongs\;V\;\;such\;that\;u + \left( { - u} \right) = {0_v}\\\\
6)ku\;belongs\;V\;\;for\;all\;k\;belongs\;F\;\;and\;u\;belongs\;V\;\left( {F = reals\;Or\;complex} \right)\\\\
7)k\left( {u + v} \right) = ku + kv\;,\;k\;belongs\;F\;,\;u,v\;belongs\;V.\\\\
8)\left( {{k_1} + {k_2}} \right)u = {k_1}u + {k_2}u\;\;\;,\;{k_1},{k_2}\;belongs\;F\;,\;u\;belongs\;V\\\\
9){k_1}\left( {{k_2}u} \right) = \left( {{k_1}{k_2}} \right)u\\\\
10)1u = u\\
\end{array}\]
