**III.5 Buildings**

*Mark Ronan*

The invertible linear transformations on a vector space form a group, called the *general linear group.* If *n* is the dimension of the vector space and *K* is the field of scalars, then it is denoted by GL_{n} (K), and if we pick a basis for the vector space, then each group element can be written as an *n* × *n* matrix whose DETERMINANT [III.15] is nonzero. This group and its subgroups are of great interest in mathematics, and can be studied “geometrically” in the following way. Instead of looking at the vector space *V*, where of course the origin plays a unique role and is fixed by the group, we use the PROJECTIVE SPACE [I.3 §6.7] associated with *V*: the points of projective space are the one-dimensional subspaces of *V*, the lines ...