Question

The vector space which has only additive identity element zero

Answer 1

A vector space under the group of addition
1. if A and B are vectors then, (A + B) is a vector.

2.for vectors A , B and C, (A + B ) + C = A + (B + C)

3. if u is the vector and then , u + 0 = 0 + u = u , where 0 is the zero vector. hence, according to definition of additive identity . zero vector is defined to be an additive identity .

4. if a is the vector then, -a is the inverse of a
then, a + (-a) = (-a) + a is true e.g., additive inverse.

In Group , we can easily show that an additive identity is unique. Thus, the zero vector is the only additive identity .