How to show that the set of functions $1,x,x^2,x^3...$ is linearly independent? Show that the set of functions $1,x,x^2,x^3...x^n...$ is linearly independent on any interval $[a,b]$.
If $$c_1+xc_2+x^2c_3+x^3c_4...=0$$ we should show $$c_i=0,\quad i=1,2, \ldots$$
how could I start?
My second question: is it linearly independent on $C[0,1]$?

Consider the polynomial $p(x)=c_1+c_2x+\dots+c_nx^{n-1}$. How many zeros can a polynomial of order $n-1$ have? How many zeros does your polynomial $p$ have?
