Definition:Transpose of Linear Transformation

Definition
Let $R$ be a commutative ring.

Let $G$ and $H$ be $R$-modules.

Let $G^*$ and $H^*$ be the algebraic duals of $G$ and $H$ respectively.

Let $\mathcal L_R \left({G, H}\right)$ be the set of all linear transformations from $G$ to $H$.

Let $u \in \mathcal L_R \left({G, H}\right)$.

The transpose of $u$ is the mapping $u^t: H^* \to G^*$ defined as:
 * $\forall y' \in H^*: u^t \left({y'}\right) = y' \circ u$

where $y' \circ u$ is the composition of $y'$ and $u$.

The transpose $u^t: H^* \to G^*$ is itself a linear transformation.