MHB How to Prove the Lagrange Interpolation Formula?

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The discussion centers on proving the Lagrange interpolation formula, specifically showing that the sum of products of a polynomial evaluated at nodal points and the Lagrange basis functions equals the polynomial itself. Participants express uncertainty about how to initiate the proof, indicating familiarity with Lagrange polynomials but lacking clarity on the proof's structure. Key points include the need to understand the properties of Lagrange basis functions and their role in polynomial interpolation. The conversation emphasizes the importance of establishing the relationship between the polynomial and its interpolating form. Overall, the thread seeks guidance on effectively starting the proof process.
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$\text{Let } L_{n,i}, i = 0,...,n, \text{be the Lagrange nodal basis at} x_0 < x_1<...<x_n$. Show that, for any polynomial $q \in P_n$

$$\sum_{i=0}^nq(x_i)L_{n,i}(x)= q(x)$$

I don't know how to begin this proof. I know what a lagrange polynomial is, but I am not sure how to begin. If someone could give me a point to start please.
 
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