Question

Consider the following functions. f1(x) = 0, f2(x) = x, f3(x) = ex g(x) = c1f1(x) + c2f2(x) + c3f3(x) Solve for c1, c2, and c3 so that g(x) = 0 on the interval (−[infinity], [infinity]). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.) {c1, c2, c3} =

Answer 1

There are infinitely many solutions for any choice of $c_1$ and $c_2=c_3=0$.

$x$ and $e^x$ are linearly independent and the only way to get $c_2f_2+c_3f_3=0$ is to choose $c_2=c_3=0$.