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Let ##X = \{x_1 , \dots , x_n\}##. Then ##\text{aff}(X) = \text{intersection of all affine spaces containing X}##. Let ##C(X)## be the set of all affine combinations of elements of ##X##. We want to show that these two sets are equal. First we focus on the ##\text{aff}(X) \subseteq C(X)## inclusion. If we can show that ##C(X)## is affine and that it contains ##X## then this inclusion will hold. Let's just say that it's obvious that it contains ##X##. So I want to prove that ##C(X)## is affine. Here is where I hit a roadblock. In my reference it only states that an affine space is a translate of a linear subspace. This is really the only definition I am given. Does this mean that I have to show that ##C(X)## is the translate of some linear subspace?