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Let \(x_1, \dots, x_n \in \mathbb{K}\) be pairwise distinct and \(L_1, \dots, L_n\) the associated Lagrange polynomials. Show that every \(P \in \mathbb{K}_{n-1}[X]\) writes uniquely as $$ P = \sum_{i=1}^{n} c_i \, L_i \quad \text{with } c_i \in \mathbb{K}, $$ and that the coefficients are necessarily \(c_i = P(x_i)\). Hint. For uniqueness, evaluate at \(x_j\) and use the Kronecker property.
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