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Concept
Affine combination
Summary
In mathematics, an affine combination of x1, ..., xn is a linear combination
: \sum_{i=1}^{n}{\alpha_{i} \cdot x_{i}} = \alpha_{1} x_{1} + \alpha_{2} x_{2} + \cdots +\alpha_{n} x_{n},
such that :\sum_{i=1}^{n} {\alpha_{i}}=1. Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients \alpha_{i} are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the \alpha_{i} are elements of K (or \mathbb R for a Euclidean space), and the affine combination is also a point. See for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span. The affine combinati
such that :\sum_{i=1}^{n} {\alpha_{i}}=1. Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients \alpha_{i} are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the \alpha_{i} are elements of K (or \mathbb R for a Euclidean space), and the affine combination is also a point. See for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span. The affine combinati
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