I'm reading "Lectures on Polytopes" by Gunter Ziegler.
The author first introduces the components of oriented matroids in affine case,
then making a transition to linear case, with the condition "1z = 0" embedded into the matrix V.
In the subsequent discussion, the author seems to use only the linear notions, such as in this one:
However, this is just my guess. I'm not sure if it is the case. Is it correct to assume that the matrix V in Definition 6.5. has only 1's in its first row? If not, please explain why the condition "1z = 0" is no longer necessary. All the theory the author's built up up to this point have this condition.