[BeetleB]

> '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).

I'm not sure how Terence meant it, but for some people, it actually means that some property that holds for f(x) will also hold for f(ax + b).

> So then saying 'suffices to check basis elements' means 'just check f(x), and the others all fall out for free'.

Yes, but your statement confuses me more than Terence's :-)

A given vector space has basis elements (e.g. x, y and z unit vectors for 3-D Cartesian space). It means that if you can show the property is true for the basis elements, you've now shown it's true for any vector in that space. One needs to show linearity holds to assume this.

> 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.

That actually helped - thanks.