Dual map

Definition

Let $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \colon V \to W$ be a linear map. The dual map of $T$, also called its transpose, is the map $$T^* \colon W^* \to V^*, \qquad T^*(f) = f \circ T,$$ between the dual spaces.

Remarks

The dual map is linear. It reverses the order of composition: $(T \circ S)^* = S^* \circ T^*$, and $(\id_V)^* = \id_{V^*}$. In dual bases, the matrix of $T^*$ is the transpose of the matrix of $T$, see the matrix of the dual map. The kernel of $T^*$ is the annihilator of the image of $T$.

Explanations

L1 human

A linear form on $W$ can be pulled back to $V$: first apply $T$, then the form. The dual map does this pullback. It goes in the opposite direction, from $W^*$ to $V^*$.