Set of Linear Transformations is Isomorphic to Matrix Space/Corollary

Corollary to Set of Linear Transformations is Isomorphic to Matrix Space

Let $R$ be a commutative ring with unity.

Let $M: \struct {\map {\LL_R} G, +, \circ} \to \struct {\map {\MM_R} n, +, \times}$ be defined as:

$\forall u \in \map {\LL_R} G: \map M u = \sqbrk {u; \sequence {a_n} }$

Then $M$ is an isomorphism.

Proof

Follows directly from Set of Linear Transformations is Isomorphic to Matrix Space.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices: Theorem $29.2$