Proof of all polynomials with degree n can for a basis

Question

Prove that {$1, x, x^2, x^3, ... , x^n$} form a basis of $P_n$ ($n$ is a non-negative integer) , the space of the polynomials of degree $n$. Using a general proof.

Answer 1

You need to show two things, first that they are linearly independent and second that they generate the set of all polynomials.

I'll give you a hint for linear independence:

You want to show that if for all $x \in \mathbb{R} $

$$\alpha_0 + \alpha_1 x + \alpha_2 x^2 + \dots + a_nx^{n} = 0 \tag{A}$$

then $\alpha_0 = \alpha_1 = \dots = \alpha_n = 0$.

To do that, note that the polynomial $(\mathrm{A})$ is $0$ for all $x \in \mathbb{R}$. However the zero-polynomial

$$0 + 0 \cdot x + 0 \cdot x^2+ \dots + 0 \cdot x^n = 0 $$

is also $0$ for all $x \in \mathbb{R}$.