I'm learning Linear Algebra and Convex Optimization simultaneously, I notice that the affine hull is, to some extent, analogous to the span, but when I read the lines "We define the affine dimension of a set C as the dimension of its affine hull" and "As an example consider the unit circle in $\mathbb{R}^2$, $\{x \in \mathbb{R}^2 \mid x_1^2 + x_2^2 = 1\}$. it's affine hull is all of $\mathbb{R}^2$, so its affine dimension is $2$." I'm wondering why the dimension is 2.

What's the exact definition of "The dimension of the affine hull"



If $C \subseteq \mathbb{R}^n$, from optimization we know that,

$$ aff(C) = S + C $$

where $aff(C)$ is affine hull of $C$, and $S$ is a unique linear subspace in $\mathbb{R}^n$ parallel to $aff(C)$. If $aff(C)$ contains the origin, then, $aff(C)$ is itself a subspace which is the case in your circle example.

Therefore, dimension of $aff(C)$ is defined as the dimension of $S$.

  • $\begingroup$ I'm a beginner, would you please give me some suggestions about how to learn optimization well. Some recommended courses, book etc. Thanks a lot! $\endgroup$ – When Apr 30 '15 at 1:26
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    $\begingroup$ I'm also a beginner in optimization :). However, my suggestion is to try to visualize things as much as possible. Wikipedia usually provides good insights about abstract concepts. $\endgroup$ – Mehdi Jafarnia Jahromi Apr 30 '15 at 1:30
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    $\begingroup$ "Convex Optimization Theory" and "Convex Optimization Algorithms" by Dimitri Bertsekas are both excellent books that complement each other. You would want to read the former first. $\endgroup$ – ChrKroer Apr 30 '15 at 1:51
  • $\begingroup$ Really, really appreciate it! $\endgroup$ – When Apr 30 '15 at 2:38

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