I got stuck on showing that the set $\displaystyle S=\{ x^1, x^2,...,x^n\}$ is linearly independent in the continuous function space. Suppose there exist $\displaystyle a_1,a_2,...,a_n$ such that $\displaystyle a_1x_1+a_2x^2+...+a_nx^n=0$ for all x, I need to show that $\displaystyle a_1=a_2=...=a_n=0$. From here I am not sure how to proceed. I'm tempted to say that the fundamental theorem of algebra says that the above equation has exactly n solutions, but this is true for all $\displaystyle x$, which is infinite, so somehow it must be the case that $\displaystyle a_1=...=a_n=0$

Any help is appreciated.