Prove that the additive identity of a vector space is unique.

Answer with explanation:

⇒The Meaning of Additive Identity is that element in a set when added to any member or elements of set yields that element.

Consider a system of forces under a set F, which is a vector space under a field.

The set F having three forces a, b, c will be a vector space if,

1.→ a + (b+c) = (a+b) + c

2.→ a+b=b+a

3.→There must exist a force equal to 0, such that when any of the forces added to 0 the resultant is force itself.

That is, a+0=a=0+a

→→So, If you consider any vector space over a field, there exist an element in that set equal to 0, when added to any member of the elements of the set the result being that element. So, element 0 is the Additive Identity of any vector space.