The following is taken from a text which tries to proves a relation between dimensions of subspaces.

My first question comes from

the canonical injections $in_1:Mto Moplus N$

in line 3 and 4. My understanding of direct sum $Moplus N$ is that, when we write down this notation, $M$ and $N$ must be independent, i.e., $Mcap N=0$. But since $M$ and $N$ are two arbitrary subspaces as indicated in line 1 an 2, they are not necessarily independent. What does the author mean here? Does $oplus$ mean something other than direct sum? Furthermore, If we assume $Moplus N$ denotes direct sum, problems comes out immediately. In line 4, the author wrote

injections $f:Mcap Nto Moplus N$

If $Mcap N=0$, then the function $f$ is just the trivial $0to 0$. So is the following $g$. I believe this is not what the author intended to define.

The second question arises from the third line if Lemma 2.14. I don’t really know what that line of symbols and “a short exact sequence” in the subsequent line mean. Does that line of arrows mean something arcane in the circle of math? Of course, the question may due to I don’t understand what $f$ means in the first place. I wish I could understand what the author wanted to say, after I have tried hard, so could you please help me understand all these stuff? Thanks a lot.