Index
We introduced linear algebra lingo in Problem 7 where we saw that a line, along with the points incident with that line, formed a subspace. Similarly any single point (no lines) formed a subspace. It would be nice if we could come up with a reasonable defintion of dimension so that the dimension of a line (which is uniquely determined by 2 points for linear spaces) is 1 and that of a point (uniquely determine by the 1 point) is 0. We can in fact do this in a natural way as follows. Given a near-linear geometry (P, L) we have the following definitions:
Definition. The closure cl(X) of a subset X of P is defined to be the the smallest subspace of (P, L) that contains X. We will call X a generating set for the subspace cl(X) and we say that X generates cl(X).
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Definition. Given a subspace S of a near-linear geometry (P, L), we define the dimension of S to be 1 greater than the smallest number of elements in a generating set for S. More formally,
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Of course this last definition begs that the following question to be answered: Does every subspace S have a generating set? Before going on please convince yourself that it does. What then would be the generating set of the subspace  , and what would be the dimension of the empty set?
We seem to be going backwards in approaching our definition of dimension from the path that was taken in linear algebra, where we first discovered the ideas of linear independence and dependence before moving on to bases and dimension. But no matter. It might seem reasonable to consider any generating set X for S which also has the fewest number of elements, that is cardinality(X) = dim S, to be akin to a basis for S. If that is so, would all bases for S have the same cardinality? For the near-linear space S below
X = {1,2,3} generates this geometry, but then so too does Y = {4, 5, 6, 7}. Note that if we remove any point from either of these generating sets, then that point no longer lies in the closure of the set that remains. For example,
and similarly for Y. It seems that both X and Y are linearly independent and both span or generate S, but that they do not have the same number of elements! So we cannot simply define dim S to be the number of elements in a basis (that is, in a linearly independent and generating set). More formally,
Definition. A subset X of P is said to be linearly independent if for any  , then  . Otherwise X is said to be linearly dependent. A linearly independent set which generates a near-linear geometry will be called a basis for that geometry.
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Can you see that for the example above, the set X = {1,2,3,4} is linearly dependent?
Is the empty set always linearly independent? What about a subset consisting of just one point? Of just two points?
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PROBLEM 11: (a) Show that cl(X) equals the intersection of all the subspaces containing X. Solution
11: (b) Find a near-linear geometry that is not isomorphic to the example above that has two bases of different cardinalities. Solution
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References: Batten, Batten & Beuteslspacher, Bennett, Beutelspacher & Rosenbaum.
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