Prove linear independence of functions $1,x,x^2,...,x^n$

Question

I need to prove that the set of functions $\{1,x,x^2,\dots ,x^n\}$ is linearly independent in the space of continuous functions $С[0,1]$ for any value of $n\in\Bbb N$.

I know that I can do this using mathematical induction, but is there a simpler way? If there is can you please show one, thanks in advance.

Answer 1

Let $$ C_0+C_1x+C_2x^2+...+C_nx^n=0$$ Let $x=0$, to get $C_0=0$

Differentiate and let $x=0$ to get $$C_1=0$$

Differentiate again and let $x=0$ to get $$C_2=0$$

and so forth until you come up with $$ C_0=C_1= C_2=...=C_n=0$$

Thus your set is linearly independent.

Answer 2

Choose $n$ distinct reals $x_k$ in $(0,1)$ and form the system of equations

$$C_0+C_1x_k+C_2x_k^2+\cdots C_nx_k^n=0$$ for $k=0$ to $n$.

The determinant of the system is of the Vandermonde form and its value is the product of all $x_i-x_j$, hence nonzero.

Hence we only have the trivial solution.