• V is closed under addition
• V is commutative
• V is associative
• For every A there is a zero vector such that [V+0=V]
• For all Vectors in V there is a -V such that (-V)+V=0

And of a Scalar Multiplication such that:

• V is closed under Scalar Multiplication
• For All r,s are a set of real numbers and V is a Vector then (r+s)V= rV+sV
• r is a set of real numbers and V, W are vectors, r(V+W)= rV+rW
• r,s are a set of real numbers and V is a vector, (rs)V=r(sV)
• 1*V=V

Proofs:

• Two vector after adding are still a vector, the commute a because they have same number of column array.
• The order of adding the vectors doesn’t affect the result
• (V+W)+U=V+(W+U) the parenthesis doesn’t affect the result
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