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

Let $u \in \map {\LL_R} {G, H}$.

The transpose of $u$ is the mapping $u^\intercal: H^* \to G^*$ defined as:

$\forall y \in H^*: \map {u^\intercal} y = y \circ u$

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

Also see

• Definition:Transpose of Matrix

• Transpose of Linear Transformation is a Linear Transformation

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations