Main Idea
Solving an interpolation problem leads to a problem in linear algebra where we have to solve a matrix. Using a standard monomial basis for our interpolation polynomial we get the Vandermonde matrix. By choosing another basis, the Lagrange basis, we get the much simpler identity matrix = δi,j which we can solve instantly: the Lagrange basis inverts the Vandermonde matrix.
This construction is the same as the Chinese Remainder Theorem. Instead of checking for remainders of integers modulo prime numbers, we are checking for remainders of polynomials when divided by linears.
Read more about this topic: Lagrange Polynomial
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