Question: Let V,W be vector spaces over field F. We mark L(V,W) as the vector space of linear transformations from V to W. Let $v_0 \ne 0$. We define a transformation: $\Psi: L(V,W) \to W$ that sends a linear transformation $T \in L(V,W)$ to $T(v_0) \in W$
What is the image of $\Psi$ ? Evaluate $dim(Ker \Psi)$
What I did: I said that $dim(V)=m, dim(W)=n$, therefore $dim L(V,W)=mn$ I see that $\Psi$ can't be injective because the dimensions of L and W are not equal. $Im\Psi=\Psi(T)=T(v_0)$ Can't figure how should I calculate the dim of the Kernel. A thought I had is that since dimW < dimL then if $\Psi$ is surjective then $dimKer\Psi = mn-m$, but I don't know what if it's not surjective.
Thanks in advance