# Two meanings to affine independence? (help me clear up my misunderstanding)

I must be misunderstanding something. Let's look at the following two definitions for a set of points $S=\{v_1,v_2,...,v_k\}$ to be affinely independent:

1) S is affinely independent if the set $\{v_2-v_1, v_3-v_1, ..., v_k-v_1 \}$ is linearly independent. see http://homepages.rpi.edu/~mitchj/handouts/faces/.

2) If no three points in $S$ lie on a line, no 4 points lie on a plane, no 5 points lie on a 3-dimensional subspace, etc. See Affine dimension of a simplex

Here is an example to show that the two definitions are inequivalent: take one facet of a cube. If we translate it such that it includes the origin, we get that it has affine dimension 2 according to definition 1.

However, if we take three vertices from that facet, we find that they do not lie on a line, and therefore they are affinely independent. Hence the affine dimension of the facet is 3.

Thanks.

• If $a_1,...,a_n$ are ai., then the dimension of the affine hull is $n-1$ (the dimension of the correspinding linear space). There is no contradiction above. I prefer the definition that $a_k$ are ai. iff $\binom{1}{a_k}$ are linearly independent. – copper.hat May 20 '15 at 19:21
• Thank you! misconception cleared. – JQX May 28 '15 at 16:35