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Let $k$ be a field and $\phi: V \to V$ a linear transformation over $k$. I want to show that $\phi^*, \phi_*: L(V,V) \to L(V,V)$ given by $$\phi_*(\psi)=\phi \circ\psi$$ $$\phi^*(\psi)=\psi \circ\phi$$ are linear transformations.

The first one seems pretty obvious: $$\phi_*(\psi_1+\psi_2)=\phi \circ(\psi_1+\psi_2)=\phi(\psi_1+\psi_2)= \phi(\psi_1)+\phi(\psi_2)=\phi_*(\psi_1)+\phi_*(\psi_2)$$ by linearity of $\phi$.

I have no idea how to approach the second one, though, since we don't know anything about the linearity of $\psi$. Any hints?

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    Is $L(V,V)$ the space of all linear transformations from $V$ to $V$? If so, we know $\psi$ is linear. In this case, $\phi^*(\psi_1+\psi_2) = (\psi_1+\psi_2)(\phi) = \psi_1\circ \phi + \psi_2 \circ \phi = \phi^\star(\psi_1) + \phi^\star(\psi_2)$. – 2017-01-26
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    @walkar Absolutely right, got lost in notation. Thanks! – 2017-01-26

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