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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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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 GLn (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 ...

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