I'm trying to solve this exercise:

A subset F of an affine space is an affine subspace if and only if for all points A and B of F, the inclusion <A, B> ⊂ F holds.

However, i don't understand what <A, B> is intended to represent in this context, can you explain me ?


  • 2
    $\begingroup$ It's probably the line containing $A$ and $B$ (a one dimensional affine subspace). The notation should be defined somewhere earlier in the text. $\endgroup$ – Ethan Bolker Apr 18 at 13:09

For two points, the notation $\langle A, B\rangle$ means the line through those points. Explicitly, we have

$$\langle A,B \rangle = \{tA+(1-t)B : t \in \mathbb{R}\}$$

  • $\begingroup$ Maybe I'm misunderstanding. How would you explicitly give the affine hull of two points? Also what is the definition of an affine space? Is it a translated vectorspace? That is certainly equivalent to $A,B \in F \implies \langle A, B \rangle \subseteq F$ as I have written above. $\endgroup$ – Xabu Apr 19 at 2:53
  • $\begingroup$ My apologies. I was misled by the word "connecting", and didn't notice that your definition had $\ t\in\mathbb{R}\ $ rather than $\ t\in [0,1]\ $. I should have waited till I was properly awake before trying to comment. I have now made a minor edit to your answer and changed my downvote to an upvote. $\endgroup$ – lonza leggiera Apr 19 at 6:16

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