Been working on this assignment for hours now and I got to a point where I have a couple of questions left and I just can't get anywhere with them... One of the questions is as follows:
Let $U, V, W$ be three finite-dimensional $K$-Vectorspaces with $\dim W \ge 1$. Let $\alpha \in L(U, V)$ be a specifically chosen linear transformation. Using this we define the Pullback (What does Pullback even mean? Never had this anywhere in my lecture) of linear transformations:
$$\alpha^* \colon L(V, W) \to L(U, W); \quad f \mapsto f \circ \alpha$$
1) Show that, $\alpha^*$ is a linear transformation.
2) Show that, $\alpha^*$ is injective, when $\alpha$ is surjective.
3) Show that, $\alpha$ is injective, when $\alpha$ is surjective.
My first problem is that I'm unable to understand $\alpha^*$. What does the $f$ mean here? It's no where defined. I know what a linear transformation is and I know that in order to prove it, I have to show that:
(i) $\alpha^*(x + y) = \alpha^*(x) + \alpha^*(y)$
(ii) $\alpha^*(cx) = c\alpha^*(x)$ for all $x, y \in V$ and $c \in K$.
Any help would be appreciated. It's already 1:30 AM and I'm not getting any further without understanding $\alpha^*$...