Matrix Entrywise Addition forms Abelian Group/Examples/nxn Matrices over Real Numbers
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Example of Matrix Entrywise Addition forms Abelian Group
Let $\R^{n \times n}$ denote the set of order $n$ square matrices over the set $\R$ of real numbers.
Then the algebraic structure $\struct {\R^{n \times n}, +}$, where $+$ denotes matrix entrywise addition, is an abelian group.
Proof
From Real Numbers under Addition form Infinite Abelian Group, $\struct {\R, +}$ is an abelian group.
It follows from Matrix Entrywise Addition forms Abelian Group that $\struct {\R^{n \times n}, +}$ is an abelian group.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.7$