It's really the same idea. Suppose a, b and c are such that af + bg + ch =0. This is really a function equation. That is it means for all t, af(t) +bg(t) + ch(t) = 0. Now one way to prove independence is to find 3 different t values to get 3 equations in a, b and c whose only solution is 0. If you happen to know about Wronskians, there's another approach.