Abstract. In this paper we present a short proof of the Hermite's formula for polynomial interpolation using the theory of linear algebra, without using Taylor series expansion as in the classic proof of this formula. In this construction we use a schema of interpolation defined by the inverse of a matrix.Mathematics Subject Classification 2010: 65D05, 65D15, 15A09. Key words: Hermites's formula, polynomial interpolation, interpolation scheme.The Hermite's formula for polynomial interpolation is well known and it is successfully used in practice. It is described in many books of numerical analysis, e.g. [2,3,8]. Our goal is to offer a way to build the Hermite's interpolation polynomial and Hermite's formula for its representation. Based on the idea from [1], which is extended in an interpolation scheme in [4], we take a certain base in the space of polynomials of degree at most n − 1 ∈ N. Then with the interpolation scheme, we represent the Hermite interpolation polynomial and we obtain the Hermite's formula.Further, we consider a particular case of this schema. Let n ∈ N * , X be a vector space, Y a n-dimensional subspace of X and U : X → R n a linear operator. We denote by V : R n → Y an isomorphism between the two spaces. We assume that the operator U V : R n → R n is invertible.We define the operator P U : X → Y by P U = V (U V ) −1 U and we call it an interpolation operator of X by elements of Y , relative to the operator U .
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