Are the functions $f, g, h$ given below linearly independent? If they are not linearly independent, find a nontrivial solutions to the equations below:
$$f(x)=0, \quad g(x)=\cos(9x), \quad h(x)= \sin(9x)$$
My take so far is that, I have computed the Wronskian via Wolfram, and I got Wronskian as $0$, but I know these functions are linearly dependent because they are not $0$ at all $R$ but I have no idea how to find nontrivial solutions to this question. I think I should have an answer as follow:
$$C_1(0) + C_2(\cos(9x)) + C_3\sin(9x) = 0$$
Where $C$s are constant. Maybe they don't necessarily have to be constant? I'm not sure.
Could I get some help on finding those constants? I'm having a hard time finding linear combinations.
Best Answer
A collection $\{v_1,\dotsc,v_n\}$ in a vector space $V$ is linearly independent if the equation $$ \lambda_1v_1+\dotsb+\lambda_nv_n=0 $$ is only solved by $\lambda_1=\dotsb=\lambda_n=0$.
In your case, you have the collection $\{f,g,h\}$, which can be viewed as a collection in the vector space of functions $\Bbb R\to\Bbb R$.
Now, note that $$ 1\cdot f+0\cdot g+0\cdot h=f=0 $$ Do you see why this proves that your collection is linearly dependent?