---
title: "additive identity is unique in a vector space"
type: concept
related: [Commutivity]
source: https://www.jemoka.com/posts/kbhadditive_identity_is_unique_in_a_vector_space/
confidence: high
status: active
---

Assume for the sake of contradiction \(\exists\ 0, 0&rsquo;\) both being additive identities in vector space \(V\).
Therefore:
\begin{equation} 0+0&rsquo; = 0&rsquo; +0 \end{equation}
commutativity.
Therefore:
\begin{equation} 0+0&rsquo; = 0 = 0&rsquo;+0 = 0' \end{equation}
defn. of identity.
Hence: \(0=0&rsquo;\), \(\blacksquare\).
