Theorem

A vector space has a unique additive identity.

Proof

Suppose 0 and 0’ are both additive identities for some vector space V. Then \({latex.inline[0' = 0' + 0 = 0 + 0' = 0](0' = 0' + 0 = 0 + 0' = 0)} . The left equality holds beacuse 0 is the additive identity. The middle one holds because of the commutative property of the vector space. The right equality holds because 0’ is also an additive identity. Thus, we get \){latex.inline[0' = 0](0' = 0)}.