So I'm having doubts about this problem.
Let $$ V=\operatorname{span}\{2, 3\sin(x)\cos(x), \cos(2x)-1,\sin(2x)+1,\cos^2(x),\sin^2(x)\} $$ Find a basis for $V$.
I reduced it to $$V=\operatorname{span}\{3\sin(x)\cos(x),\cos^2(x),\sin^2(x)\}$$ using trigonometric identities and I don't think I can reduce it anymore, but I am not quite sure, would appreciate any help, thanks in advance.
You wish to show the functions $f(x)=3\sin x\cos x$, $g(x)=\sin^2 x$, and $h(x)=\cos^2 x$ are linearly independent. So suppose we have a linear relation $$af(x)+bg(x)+ch(x)=0$$ for some $a,b,c\in\mathbb{R}$; we wish to show that $a=b=c=0$. To show this, we can try plugging in some values for $x$. For instance, setting $x=0$, we have $f(0)=g(0)=0$ and $h(0)=1$, so we get $$0=af(0)+bg(0)+ch(0)=c,$$ so $c=0$. Setting $x=\pi/2$, we have $f(\pi/2)=0$ and $g(\pi/2)=1$, so we have $$0=af(\pi/2)+bg(\pi/2)=b.$$ So our relation is actually just of the form $af(x)=0$. Now just plug in any value of $x$ such that $f(x)\neq 0$ to conclude that $a=0$.