Vector Space has Unique Additive Identity
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Theorem
A vector space has unique additive identity.
Proof 1
Let $V$ be a vector space.
Let $0$ and $0'$ both be additive identities for $V$.
Because $0$ is an additive identity:
- $0' = 0' + 0$
From commutativity of vector space addition:
- $0' + 0 = 0 + 0'$
Because $0'$ is an additive identity:
- $0 + 0 = 0$
Hence:
- $0' = 0$
Thus $V$ has a unique additive identity.
$\blacksquare$