I'm working on a textbook problem to help my understanding of basis, but I'm really stuck on these type of problems. I understand that a basis is essentially a linearly independent set that spans a vector space, but I'm having trouble applying it to this problem.
The problem states: For each of the differential equations, determine whether the given solutions are a basis for the set of all solutions. Problem Part of what confuses me on this particular problem is e being raised to a power. How can I find if the given solutions are linearly independent in this case?
First of all, the dimension of the space of solutions is 3, and there are 3 vectors so that's okay there. Then you have to verify those vectors are indeed solutions. And finally you have to check if they are linearly independent. If so, then you have indeed a basis.
Edit: for your edit, vector 1 and 2 are linearly independent, so you just have to check that the linearity doesn't work for, let's say the second coordinate: show that for all $t\in\mathbb R$, $ae^{2t} + be^{4t} = 0$ implies $a=b= 0$ (find all $a$ and $b$ possible for $t = 1$ and $t= 2$ for instance and then conclude).