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Let $S = \{x_1, x_2 \ldots x_n\}$ be a set of $n$ points in $\mathbb{R}^d$, then how can one geometrically interpret or visualise the affine hull of $S$?

It is somewhat straightforward to think about the convex hull intuitively, but unable to get any geometric interpretation for affine hull especially when the $|S|$ exceeds the dimension of the space $\mathbb{R}^d$.

RobPratt
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Akshay Bansal
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    Think small examples: the affine hull of two distinct points is the line between them (not just the line segment). For three points that aren't colinear, it will be the unique plane passing through all three. Affine subspaces are translates of linear spaces, so the result is something closer to the span of the vectors, except the result doesn't have to pass through the origin. – Theo Bendit Oct 07 '22 at 00:03
  • "dimension-theory-algebra" refers to the lengths of chains of (prime) ideals, which are not relevant here. In general, if you aren't sure whether a tag is appropriate, it's better to omit it than include it (but no harm done in this instance). – Jacob Manaker Oct 07 '22 at 00:04
  • @JacobManaker: Apologies for the incorrect tag. – Akshay Bansal Oct 07 '22 at 00:11
  • Related, possibly helpful: https://math.stackexchange.com/questions/2262258/what-does-it-mean-to-be-affinely-independent-and-why-is-it-important-to-learn/2262272#2262272 – Ethan Bolker Oct 07 '22 at 00:19
  • Start with the convex hull of $S$, and draw all lines that pass through at least two points of the convex hull. Alternatively, translate $S$ so that $0$ is in the set and take the linear hull (and the translate the whole shebang back). – copper.hat Oct 07 '22 at 02:03

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If $S=\{x_0\}\sqcup T$, then the affine hull of $S$ is $$x_0+\operatorname{span}(T-x_0)$$ where $T-x_0=\{t-x_0:t\in T\}$ and $\operatorname{span}$ refers to the linear subspace generated by that set.

If you aren't sure why the formula I gave is the affine span, then proving from the definitions that it is so is a good exercise.

Jacob Manaker
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