Linear and multilinear algebra; matrix theory Definition

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 $T$ .

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Vector space definition
Linear map definition
Linear form definition
Space of linear maps definition
Dual space definition
Dual map definition

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Matrix of the dual map is the transpose

Tag	`0C2TQMW5A1`
URI	`m2://0C2TQMW5A1`
Kind	definition
Writing	W0 stub
MSC	15A04
Defines	Dual map
Uses	Dual space, Linear map

Dual map

Definition

Remarks

Reading path

Where this leads

Further reading

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