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I just learned what an affine space is and its basic properties. It is formally defined as $\Bbb A=(A,V,f)$ where $A$ is a set of points, $V$ is a vector space, and $f:A^2→V$ is a function st. for any $x,y,z∈A$, we have

  1. $f(x,y)+f(y,z)=f(x,z)$;
  2. $f_o (x):A→V≔f(o,x)$ is a bijection from $A$ to $V$ for any fixed $o∈A$.

Here $f(x,y)$ is also denoted as $y-x$.

I also learned that an affine set is defined as (using field $\Bbb R$ for example) the set $\{\theta x+(1-\theta)y:\theta\in\Bbb R\}$ where $x,y\in\Bbb R^n$ are fixed. This concept is introduced in the first lecture of a mathematical optimization course.

I am a bit confused now about what is the relationship between the two concepts, since they are both titled "affine"? Thank you.

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  • $\begingroup$ An affine space is just like a vector space except that it does not have an origin. An affine set is just like a linear subspace except that it might not pass through the origin. Indeed, an arbitrary linear subspace of $\mathbb R^n$ is a vector space in its own right, while an arbitrary affine subset of $\mathbb R^n$ is an affine space. $\endgroup$ – user856 Aug 1 '16 at 23:58
  • $\begingroup$ Your definition of an affine set is somewhat limited. More precisely, given $x_1,\ldots,x_k\in\mathbb{R}^n$, the affine closure of $\{x_1,\ldots,x_n\}$ is the collection of all linear combinations $\sum_{j=1}^ka_jx_j$, such that $a_1,\ldots,a_k\in\mathbb{R}$ satisfies $\sum_{j=1}^na_j=1$. $\endgroup$ – Aweygan Aug 2 '16 at 0:07

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