In my class we're learning vector spaces, and in the text book there's an example with no solution and it goes like this:
If the domain of functions $f$ and $g$ is $[-1,1]$ and if they are defined $f(x) = \arcsin\left(\displaystyle\frac{2x}{1+x^2}\right)$ and $g(x) = \arctan(x)$, then $(f,g)$ is linearly independent?
I don't know how to prove this, if I can make a linear combination of one of them using the other it's dependent, but how should I go about doing that?
Thanks in advance.
If you want to show that $f$ and $g$ are linearly independent, a very similar (but different) question was asked recently. Many techniques for solving that problem are also applicable here.
If you want to show that $f$ and $g$ are linearly dependent, you have to show that $\arcsin(2x/(1+x^2))$ is some constant multiple of $\arctan x$.
Theorem: Let $f$ be a function defined in an intrval $I$, and let $F$ be an antiderivative of $f$ in $I$. Then every other derivative of $f$ in $I$ is of the form $F+costant$.
Now use above theorem to see that: $$\arcsin\left(\frac{2x}{1+x^2}\right)= \left\{ \begin{array}{ll} -\pi-2\arctan(x) & \quad x \leq-1 \\ 2\arctan(x) & \quad -1\leq x\leq1 \\\pi-2\arctan(x) & \quad x\geq1 \end{array} \right. $$
