Invertible Matrix corresponds to Automorphism

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Theorem

Let $R$ be a ring with unity.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $G$ be an $n$-dimensional $R$-module.

Let $\map {\MM_R} n$ be the $n \times n$ matrix space over $R$.

Let $\map {\LL_R} G$ be the set of all linear operators on $G$.

Then the invertible elements of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$ correspond directly to automorphisms of $\map {\LL_R} G$.

Proof

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Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices