Legendre Polynomials form a Basis

Question

How can I show that Legendre Polynomials form a basis for all polynomials? I think it should be enough to prove that they are linearly independent but how exactly should I do that?

Answer 1

Whenever you have a family $(P_n)_{n\in\mathbb{Z}^+}$ of polynomials such that $(\forall n\in\mathbb{Z}^+):\deg P_n=n$, they form a basis of the space of all polynomials. This follows from the fact that, for each $N\in\mathbb{Z}^+$, $\{P_0,P_1,\ldots,P_N\}$ is a basis of the space of the polynomials whose degree isn't greater than $N$. Note that this space has dimension $N+1$ and that there are $N+1$ polynomials here. So, all that is needed is to prove that they are linearly independent. This follows from the fact that, if you express each $P_k$ in the basis $\{1,x,\ldots,x^N\}$, then you'll get an upper triangular matrix $A$ such that the entries of the main diagonal are all different from $0$. Therefore, $\det A\neq0$.