According to this definition of affine spans from wikipedia, "In mathematics, the affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S, or equivalently, the intersection of all affine sets containing S."
They give the definition that it is the set of all affine combinations of elements of S. I am a little confused by this for two reasons.
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For each affine combination, they require $$\displaystyle \sum_{i=0}^n \alpha_i = 1$$Why is this?
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Just as affine spaces are translations of a vector space, can I think of affine combinations as a translation of a vector?
Any clarification or further details would be appreciated!
Best Answer
Suppose $S=\{x_1, x_2\}$ is a subset of $\mathbb{R}^2$ that consists of two linearly independent vectors. Without the constraint $\sum_{i=0}^n \alpha_i=1$, the set of points of the form $\alpha_1 x_1 + \alpha_2 x_2$ would be all of $\mathbb{R}^2$. The constraint forces the linear combination to lie on the line connecting $x_1$ and $x_2$.
For general $S$, to see why this constraint forces the linear combination $\sum_{i=1}^n \alpha_i x_i$ to lie on the smallest affine space containing $S$, you can rewrite $\sum_{i=1}^k \alpha_i x_i$ as $$\left(\sum_{i=1}^k \alpha_i \right)x_1 + \sum_{i=2}^k \alpha_i (x_i - x_1).$$
When $\sum_{i=1}^k \alpha_i=1$, you can think of the above expression as the span of $\{x_2-x_1, \ldots, x_n - x_1\}$ (which is a subspace), but shifted by $x_1$ (becoming an affine space).