# Show that $x, cosx,$ and $\frac{x^2}{1+x^2}$ are linearly independent in $C(\mathbb{R})$?

I assume by $$C(\mathbb{R})$$ the question means the vector space of continuous real functions, but I'm not completely sure of that.

How might I go about formally proving this? Obviously I could say that there's no way you can get the graph of any of these functions through linear combinations of the other two, but that's not very formal. Writing out $$c_1x +c_2cosx+c_3\frac{x^2}{1+x^2} =0$$ isn't very helpful either as far as I can see. Any help is appreciated!

The equation you wrote must hold for any $$x\in\mathbb{R}$$. Put $$x=0$$ into the equation and you will get $$c_2=0$$. Then put $$x=1$$ and $$x=-1$$ to show that $$c_1=c_3=0$$.
Alternative approach: $$\frac{x^2}{1+x^2}$$ cannot be a linear combination of $$x$$ and $$\cos x$$ since the latter are entire functions, while $$\frac{x^2}{1+x^2}$$ is not. $$x$$ and $$\cos x$$ are obviously linearly independent since the former vanishes at a single point, while the latter is a periodic function.
• Thank you! Could you explain what you mean when you say $x$ vanishes at a single point? – James Ronald Apr 22 at 15:05
• @JamesRonald: the only solution of $x=0$ is $x=0$, while the solutions of $\cos x=0$ form the set $\frac{\pi}{2}+\pi\mathbb{Z}$. – Jack D'Aurizio Apr 22 at 15:06