Consider $C[0,1]$: the vector space of all continuous functions on the interval $[0,1]$. Let $S$ be a subspace of $C[0,1]$ where $S =$ the span of $\{e^x, e^{-x}\}$
does the following function: $\cos(x)$ belong to $S$? In other words, can $\cos(x)$ be rewritten as a linear combination of $e^x$ and $e^{-x}$ when working with the interval $[0,1]$?
My intuition is yes, since these functions arent discontinuous, there will always be some real numbers a and b such that satisfy the following equation:
$a\cdot e^x + b\cdot e^{-x} = \cos(x)$ for all $x$ in $[0,1]$. I just dont know how to prove that..