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## Corollary: Abelian Group of Vectors Under Addition

Let $F$ be a field. The set $F^n$ of all vectors with $n$ coordinates (which we are going to identify with the set $M_{n\times 1}(F)$ of all single-column matrices with $n$ rows over the field $F$) together with the matrix addition $”+”$ constitutes an Abelian group, i.e. $(F_n, +)$ follows the rules:

1. Associativity $u+(v+w)=(u+v)+w$ for all vectors $u,v,w\in F^n$.
2. Commutativity $v+u=u+v$ for all vectors $u,v\in F^n$.
3. The zero vector $o\in F^n$ is the neutral element: $o+v=v+o=v$ for all $v\in F^n$.
4. For all vectors $v\in F^n$ there is a vector $-v$ such that $v+(-v)=(-v)+v=o.$

| | | | | created: 2020-11-29 14:23:13 | modified: 2020-11-29 14:24:58 | by: bookofproofs | references: [8684]