2
$\begingroup$

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"

thanks!

$\endgroup$
1
$\begingroup$

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 $S$.

$\endgroup$
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ "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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.