Let $V, W$ be finite dimensional vector spaces over a field $F$. Let $\mathcal{B}_{V} = \{\mathbf{v_1, \cdots, v_n} \}$ and $\mathcal{B}_{W} = \{\mathbf{w_1, \cdots, w_m} \}$ be corresponding bases. Let $\psi: V \to W$ a linear map and let $A$ be the matrix representing $\psi$ with respect to the above bases.
(a) Let $\chi: F^n \to V$ be given by $\chi(\mathbf{x}) = \displaystyle \sum_{i=1}^{n} x_i\mathbf{v}_i$. In one line, explain why $\chi$ is an isomorphism.
(b) Prove that:
$$\mathbf{x} \in \ker(A) \iff \left (\displaystyle \sum_{i=1}^{n} x_i\mathbf{v}_i \right )\in \ker(\psi)$$
(c) Conclude that $$\dim(\ker(A)) = \dim(\ker(\psi))$$
For (a) it suffices to notice that $\chi$ is linear. Since $\dim(F^n) = \dim(V) = n$, $\chi$ is forced to be an isomorphism. For (b) I really need help. It really feels like it should not be that hard, but apparently I am missing some small step that ruins everything. Any help will be much appreciated!