Affine dimension of a simplex

Question

In Stephen Boyd's book on Convex optimization he points out that k+1 affinely independent points form a simplex with affine dimension k.

My understanding of affinely independent points is that no 3 points are in a line. So if I take 4 points no 3 of which are in a line in $R^2$ than I get a simplex of affine dimension 3.

How is it possible for a set to have dimension more than 2 in $R^2$?

Please correct me if I am wrong.

On further inspection I realized that Boyd says "affine dimension of simplex". Now simplex is a convex set and affine dimension should be defined for an affine set. Isn't that correct?

Answer 1

They're affinely independent if none of them is in the affine space spanned by the others, i.e. the smallest affine space containing the others. Three points are in a plane, so the fourth point must not be in the same plane as the first three.

Answer 2

$n$ points are affinely independent iff no $3$ of them lie on a line, no $4$ of them lie on a plane, no $5$ of them lie on a 3d subspace, and so on..

Exactly, as you considered, it is possible to have $4$ points in one plane such that no $3$ of them lie on a line (e.g. a square). But they are still affinely dependent.

A precise definition of $P_0,P_1,\ldots,P_n$ points being affinely independent is that the collection of vectors $\vec{P_0P_1},\ldots,\vec{P_0P_n}$ is linearly independent.