How Does Affine Geometry Define the Join of Subspaces?

[mma]

http://planetmath.org/encyclopedia/AffineGeometry.html" writes:

In addition, we define A\vee B to be the smallest flat in \mathcal{A}(V) that contains both A and B. By Zorn's lemma, A\vee B exists. Since A\vee B is also unique, \vee is well-defined. This turns \mathcal{A}(V) into an upper semilattice. If S_1 is the associated subspace of A and S_2 is the associated subspace of B, then \operatorname{span}(S_1\cup S_2) is the associated subspace of A\vee B.

As far as I see, this is wrong. For example, let be V=\mathbb{R}^3, S_1=\{(x,0,0):x\in \mathbb{R}\} , S_2=\{(0,y,0):y\in \mathbb{R}\}, A=(0,0,1)+S_1 and B=S_2. Then A and B are skew lines, that is, there isn't any plane containing both line. While any flat with associated subspace \operatorname{span}(S_1\cup S_2) is a plane.

Am I right? If yes, then what would be the correct statement here?

 

[micromass]

Hi mma!

Yes, as far as I can see, the last sentence is not correct. It can be corrected as follows: if A=a+S₁ and B=b+S₂, then

A\vee B=a+(S_1+S_2+span\{b-a\})

 

[mma]

Yes, this seems better, thank you!