For $V = F(\mathbb R)$ be the space of real valued functions on the real line. $S$ is: $\ S = \operatorname{Span}\lbrace\sin\theta, \cos\theta, \sin\theta\cos\theta\rbrace. $
I have to find the $\dim S $
I think that $\sin{\theta}$, $\cos{\theta}$, and $\sin\theta\cos\theta$ are linearly independent and if so then the dimension of $S$ is 3. However I am unsure how to prove this. I do know that I have to prove that $a = b = c = 0$ in the following equation: $a \sin\theta + b\cos\theta + c \sin\theta\cos\theta = 0 $ but I don't know how to proceed from here.