Definition:Transpose of Linear Transformation

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.

Proof
By definition of Evaluation Linear Transformation, $$\forall x \in G: y' \in H^*: \left \langle {x, u^t \left({y'}\right)} \right \rangle = \left \langle {u \left({x}\right), y'} \right \rangle$$.

Since we have:

$$ $$ $$ $$

and:

$$ $$ $$ $$

it follows that $$u^t: H^* \to G^*$$ is a linear transformation.