> I actually wouldn't have figured it out if he hadn't pointed out this is induction.
A lot of these other ones are not that complicated either, it's just hard to parse and sometimes requires a small bit of math knowledge, which I think is within reach if you're familiar with induction and would only take e.g. a minute for you to understand if someone explained it (ideally with a blackboard than in text).
Some of those are definitely nonsense to me. But, to put my money where my mouth is and to give some examples (as you spent the time to type that out), 'invariance under linear combinations' just means that, if something holds for f(x), then it holds for 2f(x), f(x) + 10, etc. (these are all linear combinations). So then saying 'suffices to check basis elements' means 'just check f(x), and the others all fall out for free'.
As another example, convex combinations: Google 'convex hull' for images, but basically this one is just saying just check the 'extreme elements', i.e. the 'corners' at the boundary, and everything in the middle falls out for free because we have 'invariance under convex combinations'. A convex combination here is just a some point in the middle of these extremes.
While these may not be immediately obvious when reading them, the pictures or ideas are actually sometimes quite simple.
I hope that helps.