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.