Linear mapping/Dual mapping/Functorial properties/Fact

Let ${}U,V,W$ denote vector spaces over a field ${}K$ and let

$\psi \colon U\longrightarrow V$

and

$\varphi \colon V\longrightarrow W$

be linear mappings. Then the following hold.

1. For the dual mapping, we have

   ${}{\left(\varphi \circ \psi \right)}^{*}=\psi ^{*}\circ \varphi ^{*}\,.$

2. For the identity on ${}V$, we have

   ${}{\operatorname {Id} _{V}}^{*}=\operatorname {Id} _{{V}^{*}}\,.$

3. If ${}\psi$ is surjective then ${}\psi ^{*}$ is injective.
4. If ${}\psi$ is injective then ${}\psi ^{*}$ is surjective.

Proof, Write another proof