How should I check if an uncountable set of functions is linearly independent? There is an exercise in Cheney's book which is:
Prove that the set of functions $\{ u_x : x\in \mathbb{R} \}$ is linearly independent. $u_x(t)=e^{ixt}$ where $x$ and $t$ are real numbers.
I know what to do exactly for countable sets but in uncountable sets I'm not even sure what does it mean. Should I use integral?
A set $S$ of vectors is linearly independent if there is no finite, non-trivial linear combination of them that equals zero. You can also think of this as $S$ is linearly independent if and only if every finite subset of $S$ is linearly independent.
So to start proving this, you would take $$ \sum_{i = 0}^n a_ie^{ix_it} = 0 $$ where you assume that for each $i$, $a_i \ne 0$ and arrive at a contradiction.
Note: the reason why we work with finite sums is because infinite sums do not make sense in general. You need some sort of topology that allows you to take limits. There are notions of linear independence that use countable sums (series) but unless you are told this explicitly, you can expect that linear-independence is a statement about finite sums.