Affine Set Dimensions

Why

Since every affine set is a translate of some (unique) subspace, it is natural to define the dimension of an affine set as the dimension of this subspace.

Definition

The dimension of a nonempty affine set is the dimension of the subspace parallel to it. By convention, $\varnothing$ has dimension $-1$. Naturally, the points, lines and planes are affine sets of dimension 0, 1, and 2 respectively.

If an affine set has dimension $r$, then we often call it an $r$-flat.

For any $S \subset \R ^n$, we define the dimension of $A$ to be the dimension of the affine hull of $A$.

Notation

We denote the dimension of the set $S \subset \R ^n$ by $\dim S$ We have defined it so that

\[ \dim S = \dim \aff S \]

This makes sense if $S$ is affine, since in this case $\aff S = S$.

Result

Any $r$-flat has $r+1$ affinely independent points. Each of its sets of size $r+2$ are affinely dependent.