Are the functions $f, g, h$ given below linearly independent? If they are not linearly independent, find a nontrivial solutions to the equations below:
$$f(x)=e^{2x}- \cos(9x), \quad g(x)=e^{2x}+ \cos(9x), \quad h(x)= \cos(9x)$$
My take so far is that, I know these functions are not linearly independent (meaning this is surely dependent) by Wronskian (as the Wronskian determinant isn't equal to zero), but I have no idea how to find nontrivial solutions to this question. I think I should have an answer as follows:
$$ C_1(e^{2x}-\cos(9x)) + C_2(e^{2x}+\cos(9x)) + C_3\cos(9x) = 0 \qquad \text{ where } Cs \text{ are constant } $$
Could I get some help on finding those constants? I've tried $C_1$ as $-1$, $C_2$ as $1$ and $C_3$ as $0$ but that surely cannot be the answer.
Best Answer
Those functions are a red herring.
;-)You basically have the vectors $$ v-w,\quad v+w,\quad w $$ in some vector space. These vectors belong to the subspace spanned by $v$ and $w$, which has dimension at most $2$. So a set of three vectors is necessarily linearly dependent.
How to find a nonzero linear combination is easy: $$ a(v-w)+b(v+w)+cw=0 $$ gives $$ (a+b)v+(-a+b+c)w=0 $$ so we can choose $b=-a$ and so $c=2a$. If we take $a=1$, we obtain $b=-1$ and $c=2$.