I have a question which I already think to have an answer to but, I want to double check. If we have to subspace from the same vector space then the addition is well-defined, since the addition inside a vectorspace is well defined, and the two subspaces are subsets of the vectorspace. But in case we have a subspace of the vectorspace V and a subspace of the vectorspace U and we try to define an addition, in general it does not have to be well-defined, since we cannot state anything about the addition between to different vectorspaces.
Is this correct?
Thank you in advance
Yes. As an example, let $U = \mathbb{R}^9$, and $V= M_{3\times 3}(\mathbb{R})$. Even though $U\cong V$, without invoking the ismorphism (which requires choosing a specific basis), who's to say what $$(1,0,3,0,0,0,0,0,0)+\begin{pmatrix}1 & 0 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ is equal to?