Hello, I am having some trouble proving
that the set {1,x,x^2,x^3,...x^n} is linearly independent in Pn(R).
If $\displaystyle a_0 + a_1x+a_2x^2+...+a_nx^n = \bold{0}$ where $\displaystyle \bold{0}$ is the zero function. It means that we have a polynomial always equaling 0 for any $\displaystyle x$. It means that that it must be the zero polynomial because otherwise we have a non-zero polynomial having infinitely many zero (for any x) which is impossible (it can have at most n zeros). Thus, $\displaystyle a_0,...,a_n$ have to be all zero.