Suppose we have a surjective (though not necessarily injective) linear map $\varphi:U \rightarrow V$ between vector spaces $U$ and $V$. Suppose further that $U \cong U_1 \oplus U_2$. Is it possible to decompose both the vector space $V$ into a direct sum of subspaces $V_1$ and $V_2$ and the linear map $\varphi$ into a direct sum of linear maps $\varphi_1:U_1 \rightarrow V_1$ and $\varphi_2:U_2 \rightarrow V_2$? (Here, $\varphi_1 \oplus \varphi_2: U_1 \oplus U_2 \rightarrow V_1 \oplus V_2$ may be interpreted as a block matrix.)
I believe this should be possible, as I seem to be able to prove the dual result (unless I've made a mistake somewhere). Namely, if $\varphi$ is injective but not surjective and $V \cong V_1 \oplus V_2$, then $U\cong U_1 \oplus U_2$, where $U_1$ and $U_2$ are the preimages of $V_1$ and $V_2$ under $\varphi$, respectively. Moreover, $\varphi \cong \varphi_1 \oplus \varphi_2$, where $\varphi_1$ and $\varphi_2$ are the restrictions of $\varphi$ to $U_1$ and $U_2$ respectively. There are also indications that the statement in the first paragraph should be true from other areas of mathematics, such as when considering the indecomposable representations of a Dynkin type $D$ quiver (which is, in part, why I'm looking at this). However, I'm really struggling to construct such a decomposition.
One thing I tried was to state that $U_1 \oplus U_2 \cong U / \mathrm{Ker} \varphi \oplus \mathrm{Ker} \varphi$. Thus, there exists a bijection $$ \phi=\begin{pmatrix} \phi_{11} & \phi_{12} \\ \phi_{21} & \phi_{22} \end{pmatrix}:U_1 \oplus U_2 \rightarrow U / \mathrm{Ker} \varphi \oplus \mathrm{Ker} \varphi.$$ In this way, one obtains injective maps $$ \begin{pmatrix} \phi_{11} \\ \phi_{21} \end{pmatrix}:U_1 \rightarrow U / \mathrm{Ker} \varphi \oplus \mathrm{Ker} \varphi$$ and $$ \begin{pmatrix} \phi_{12} \\ \phi_{22} \end{pmatrix}:U_2 \rightarrow U / \mathrm{Ker} \varphi \oplus \mathrm{Ker} \varphi.$$ Moreover, $U / \mathrm{Ker} \varphi \cong \mathrm{im} \varphi \cong V$ by the first isomorphism theorem, so let $\overline{\varphi}:U / \mathrm{Ker} \varphi \rightarrow V$ be such an isomorphism. I thought that in this way, one could show that $V \cong \mathrm{im}(\overline{\varphi}\phi_{11}) \oplus \mathrm{im}(\overline{\varphi}\phi_{12})$, but I can't seem to show that the intersection of these two vector spaces is trivial. (I am perhaps just very tired as I have been working for almost 12 hours now...)
Any help would be greatly appreciated!
Edit: I should clarify that in the setting I am working with, we allow for a zero direct summand in the decomposition as long as both $U_1$ and $V_1$ are not zero or both $U_2$ and $V_2$ are not zero.
No. Take $ f : \mathbb{R}^2 \rightarrow \mathbb{R}, f((x, y)) = x+y $.
Then $ \mathbb{R}^2 = \mathbb{R} \oplus \mathbb{R} $ gives a contradiction.
Edit : one other way to see this is that not every map is diagonalizable.