Hello, I'm currently self-studying "An Introduction to Convex Polytopes" and I'm having some trouble understanding the different characterizations of affine independence. I understand that for an n-family (x_{1},...,x_{n}) of points from R^{d}, it is affinely independent if a linear combination [tex]\lambda_{1}x_{1} +...+\lambda_{n}x_{n}[/tex] with [tex]\lambda_{1} + ... + \lambda_{n} = 0[/tex] can only have the value of the zero vector when [tex]\lambda_{1} = ... = \lambda_{n}=0[/tex]. This makes sense to me and I understand how it is an analogue to the definition of linear independence, which I would like to think I understand quite well.(adsbygoogle = window.adsbygoogle || []).push({});

It then states that, "affine independence of an n-family (x_{1},...,x_{n}) is equivalent to linear independence of one/all of the (n-1)-families:

[tex](x_{1}-x_{i},...,x_{i-1}-x_{i},x_{i+1}-x_{i},...,x_{n}-x_{i})[/tex].

I'm not immediately seeing this connection, if someone would be able to help explain it to me or send me a link so that I could understand it better, that would help. I have checked Wikipedia, MathWorld, etc. already.

Thank you very much,

-Caramon

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# Characterizing Affine Independence

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