13 - Dual Map

Dual Map

Let $T:V\to W$ be a linear map of vector spaces

The dual map is defined by

$$T^{\prime }:W^{\prime }\to V^{\prime },f\mapsto f\circ T$$

Note that $f\circ T:V\to W\to \mathbf{F}$ is linear, so $f\circ T\in V^{\prime }$

Linear Property of the Dual Map

Let $T:V\to W$ be a linear map of vector spaces then

$$T^{\prime }\text{ is a linear map }$$

Proof

Let $f,g\in W^{\prime }$, $\lambda \in \mathbf{F}$
For $v\in V$ then

$$\begin{array}{rl}{T}^{\prime }(f+\lambda g)(v)& =((f+\lambda g)\circ T)(v)\\ & =(f+\lambda g)(Tv)\\ & =f(Tv)+\lambda g(Tv)\\ & =T^{\prime }(f)(v)+\lambda T^{\prime }g(v)\\ & =(T^{\prime }(f)+\lambda T^{\prime }(g))(v)\end{array}$$