The default functional link for mgcv::gam is "log", so I doubt that
your theoretical understanding applies to GAM's in general. When Simon
Wood wrote his book on GAMs his first chapter was on linear models, his
second chapter was on generalized lienar models at which point he had
written over 100 pages, and only then did he "introduce" GAMs. I think
you need to follow the same progression, and this forum is not the
correct one for statistics education. Perhaps pose your follow-up
questions to CrossValidated.com
David, thank you for your advice, has the default changed for mgcv::gam?
Based upon the help pages for the version I have (1.7-27) I had thought
that the default family was gaussian() with link "identity".
In any event I will look again at Simon Woods' book and consider
CrossValidated in the future.
I may have gotten this wrong by only referring to my memory. I'm not able
to tell by looking at either ?mgcv::gam or ?gam::gam pages where I
picked up this notion.