Sum of preimages of subspaces

Question

Let $E$, $F$ be vector spaces over a field $K$.

Let $u$ be a linear map between $E$ and $F$ and $F_1,F_2,\dots, F_n$ subspaces of $F$.

It's quite easy to show that $u^{-1}(F_1)+\dots+u^{-1}(F_n) \subset u^{-1}(F_1+F_2+\dots +F_n)$

But do we have $u^{-1}(F_1+F_2+\dots +F_n) \subset u^{-1}(F_1)+\dots+u^{-1}(F_n)$ ?

I'm quite sure that the answer is no but I can't find a counterexample.

Comments :

Of course if $u$ is surjective, then the equality follows. So we need to find some subspaces such that $F_i \not \subset \text{im} u$... I found no counterexample between $K^n$ to $K^p$...

Any ideas ?

Answer 1

Here is a counterexample.

Let $E:=\mathbb{R}$, $F:=\mathbb{R}^2$, and $u:E\to F$ be the linear map $t\mapsto (t,t)$.

Let $F_1=\{(x,0)\}$, $F_1=\{(0,y)\}$.

Then $u^{-1}(F_1)=\{0\}$, $u^{-1}(F_2)=\{0\}$ but $u^{-1}(F_1+F_2)=u^{-1}(F)=E$.

[This would work for any field, not just $\mathbb{R}$.]