A vector space is defined as having a zero vector, that is, a vector v such that for any other vector w, v + w = w.
Saying the zero vector is unique means that only one vector has that property, which we can prove as follows. Assume that v and v’ are zero vectors. Then v + v’ = v’ (because v is a zero vector). But also, v + v’ = v’ + v = v, where the first equality holds because addition in a vector space is commutative, and the second because v’ is a zero vector. Since v’ + v = v’ and v’ + v = v, v’ = v.
We have shown that any two zero vectors in a vector space are in fact the same, and therefore that there is actually only one unique zero vector per vector space.
A vector space is defined as having a zero vector, that is, a vector v such that for any other vector w, v + w = w.
Saying the zero vector is unique means that only one vector has that property, which we can prove as follows. Assume that v and v’ are zero vectors. Then v + v’ = v’ (because v is a zero vector). But also, v + v’ = v’ + v = v, where the first equality holds because addition in a vector space is commutative, and the second because v’ is a zero vector. Since v’ + v = v’ and v’ + v = v, v’ = v.
We have shown that any two zero vectors in a vector space are in fact the same, and therefore that there is actually only one unique zero vector per vector space.