I'm trying to understand this section of Weibel's An Introduction Homological Algebra, Section 1.4.
First of all, is it just a general fact about vector spaces that whenever $V=A\oplus B$, then $B\simeq V/A$? I'm also confused about why kernel of $ds+sd$ is $H_n'$. And then in the next line, it says the kernel of $ds+sd$ is the trivial homology complex.
Also still don't get why the condition $d=dsd$ has to do with the notion of a complex being split.
I'm new to this and I'd appreciate it if someone could flesh out the details.