dimension of inverse image Let $f : V \to W$ be a linear transformation between two vector spaces. Let $E$ be a subspace of $V$.
Assume $V$ to be finite dimensional. Is there a way to obtain the dimension of $f^{-1} (f(E))$ in terms of the dimension of $E$, the dimension of $ \mathrm{im}f$ and the dimension of $\mathrm{ker}f$ ?

One has $f^{-1}(f(E))=E+\ker(f)$, which sum need not be direct, so the general formula applies
$$
  \dim(f^{-1}(f(E)))=\dim(E+\ker(f))=\dim(E)+\dim(\ker f)-\dim(E \cap\ker(f)).
$$
So apart from the dimension of $E$ and of the image of $f$ (from which you can deduce that $\ker f$), you need to know hoe $E$ intersects $\ker f$.
