1. The problem statement, all variables and given/known data Let V = {f: [tex]\mathbb {R}\rightarrow\mathbb {R}[/tex]} be the vector space of functions. Are f_{1} = e^{x}, f_{2} = e^{-x} (both [tex]\in[/tex] V) linearly independent? 2. Relevant equations 0 = ae^{x} + be^{-x} Does a = b = 0? 3. The attempt at a solution My first try, I put a = e^{-x} and b = -e^{x}. He handed it back and told me to try again. I think the problem was that my a and b were not constants. But how to prove that there are no constants that will make the equation 0? I wrote some stuff down about the fact that, if a=0, then b = 0 (and the converse). Is that sufficient or am I way off?
One way to do it would be a proof by contradiction. Suppose there are constants, not both 0, a and b, such that aexp(x) + bexp(-x) = 0 for all x. Then aexp(x) = -bexp(-x), so -a/b * exp(2x) = 1 for all x. I won't complete it for you, but look at x=0, and see what restriction it places on a/b. Then look at a different point and you will reach a contradiction. (Also note, I've implicitly assumed b is nonzero, so you should handle that case as well)
how about a.e^{x}+b.e^{-x} = 0 multiply by e^{-x} a.e^{2x}+b = 0 this is not true in general for all x in the reals unless a=b=0
Yes, a = b = 0 is one solution, and is always a solution regardless of whether these functions are linearly dependent or linearly independent. The real question is whether this solution, the trivial solution, is the only solution. If so, the functions are linearly independent. If not, they are linearly dependent.