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Affinely Independent Vectors
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Barycentric Coordinates

Defining result

If $M = \aff\set{b_0, b_1, \dots , b_m}$ then for each $x \in M$ there exists $(\lambda _i)$ such that

\[ x = \lambda _0 b_0 + \lambda _1 b_1 + \cdots + \lambda _m b_m \]

with $\sum_{i} \lambda _i = 1$.
The $(\lambda _i)$ are unique if the set of vectors is affinely independent.

The barycentric coordinates for a vector $x$ in the affine hull of a set of affinely independent vectors is the sequence of unique coefficients expressing the vector as an affine combination of the set of vectors.

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