prove that the function g(x)=x does not belong in W=sp{1,cosx,sinx} .

Question

I am stuck on this question.I know that it does not belong since W does not have an x and cannot produce an x from the span of {1,cosx,sinx} however i have no idea how to prove it . Thanks

Answer 1

Suppose $x=a+b\sin(x)+c \cos(x)$ for all $x$.

$x=0 \implies a+c=0$

$x=\pi \implies a-c=\pi$

$x=\frac{\pi}{2} \implies a+b=\frac{\pi}{2}$

Hence, if such $a,b,c$ existed, we'd have $$a=\frac{\pi}{2}, b=0, c=-\frac{\pi}{2}$$

However $x=\frac{\pi}{4} \implies \frac{\pi}{4}=a+b\frac{\sqrt{2}}{2}+c\frac{\sqrt{2}}{2}$ and plugging in the vlaues of $a,b,c$ gives a contradiction