Dimension of Set of Linear Transformations

Theorem
Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module such that $\map \dim G = n$.

Let $\struct {H, +_H, \circ}_R$ be a unitary $R$-module such that $\map \dim H = m$.

Let $\map {\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$.

Then:
 * $\map \dim {\map {\LL_R} {G, H} } = n m$

where $\map \dim {\map {\LL_R} {G, H} }$ denotes the dimension of $\map {\LL_R} {G, H}$.

Proof
Consider the set:


 * $B = \set {\phi_{i j}: i \in \closedint 1 n, j \in \closedint 1 m}$

From Basis for Set of Linear Transformations, $B$ is a basis for $\map {\LL_R} {G, H}$.

It is seen that by construction, $B$ has $n m$ elements.

The result follows by definition of dimension.