Determine if set is linearly independent in folowing vector space

Question

$\{sin x,cos^2 x, cos 2x\} $ in $\mathbb R^\mathbb R $ over $ \mathbb R$

If i understand linear independence correctly, set is not lin. independent, because $cos 2x$ can be expressed by $cos^2 x-sin^2 x$, despite this I'm not sure if its enough to say that set is linearly dependent.

Answer 1

Though I assume you just made a notational mistake, it is not true that $\cos 2x = \cos^2 x - \sin x$. The correct formula is $\cos 2x=\cos^2 x−\sin^2 x$. Nevertheless, this formula will not help you since you consider $\mathbb{R}^{\mathbb{R}}$ to be a real vector space, meaning that you cannot multiply the elements of the space. You cannot multiply two vectors in a given vector space until you define some other structure which makes it more then a vector space. Thus, since $\cos 2x $ cannot be expressed using a finite linear combination of $\cos^2 x$ and $\sin x$, the set is linearly independent.

Answer 2

Suppose there are three scalars $\;a,b,c\in\Bbb R\;$ s.t.

$$a\sin x+b\cos^2x+c\cos2x=0\implies\begin{cases}a-c=0\;,\;\;&x=\frac\pi2\\{}\\ b+c=0\,,\;\;&x=\pi\\{}\\ -a-c=0, ,&x=-\frac\pi2\end{cases}$$

solving the above linear system we get $\;'a=b=c=0\;$ and thus the set is lin. ind.