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Real Affine Sets and Subspaces
Real Subspace Dimensions
Real Affine Hulls
Affinely Independent Vectors
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Real Affine Set Representations
Real Hyperplanes
Real Polyhedra
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Affine Set Dimensions


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.


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 $A$.


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 $\aff S = S$.


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