Hey I have problems with this exercise Let $B = (\sin, \cos, \sin \cos, \sin^2, \cos^2)$, and let $V = \operatorname{Span}(B)$ be a subset of $Abb(\mathbb{R}, \mathbb{R})$. Show that $B$ is a basis of $V$.
So what I need to prove is that the vectors of $B$ are linear independent.
$a\sin+ b\cos+ c\sin \cos+ d\sin^2+ e\cos^2=0 \Leftrightarrow a=b=c=d=e=0$
I tried to prove different cases.
[I EDITED MY ANSWER]
We have here that $b+e=0$,and so we have $b=-e$.
If we take $\pi=x$ we have $-b+e=0$ wwhich means $b=e \Rightarrow b=e=0$
If we take $\pi/2=x$ we have $a+ d=0$ so $a=-d$
If we take $3\pi/2=x$ we have $-a+d=0$ so $a=d \Rightarrow a=d=0$
From this follows that $c=0$ and our assumption.
Everything right now?