By Ralph Stocker, Heiner Zieschang
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Additional info for Algebraische Topologie. Eine Einfuhrung, 2. Auflage
We also must have (vi , vi ) = 0 since otherwise vi is orthogonal to each element of the basis v1 , . . , vn , so we can write wi = 1 (vi , vi ) vi and obtain an orthonormal basis. With respect to this basis, β is diagonal so if R is the invertible matrix defining the change of basis, RT β R = diag(λ1 , . . , λn ) and RT R = I. Putting P = QR we get the result. ✷ Now let us try and set this in a geometric context. Let A, B be symmetric bilinear forms on a complex vector space V which define different quadrics Q and Q .
0 ∗ ∗ ∗ ... ∗ ∗ ∗ ∗ ... ∗ ∗ ∗ ∗ ... ∗ ∗ ∗ ∗ ... ∗ and we can unambiguously choose a representative for T by taking T00 = 1. The action of the subgroup of GL(n + 1, F ) defined by these matrices on P n (F )\P n−1 (F ) which we identify with F n as usual by (x1 , x2 , . . , xn ) → [1, x1 , . . , xn ] is of the form x → Ax + b where A ∈ GL(n, F ) and b ∈ F n . The group of such transformations is called the affine group A(n, F ). Example: The simplest affine situation is the real affine line, which is just R with the group of transformations x → ax + b with a = 0.
A plane in Q defined by a point X ∈ P (V ) like this is called an α-plane. There are other planes in Q: Proposition 21 Let P (W ) ⊂ P (V ) be a plane. The set of lines ⊂ P (W ) corresponds to the set of points L ∈ Q which lie in a fixed plane contained in Q. 51 A plane of this type contained in Q is called a β-plane. Proof: We just use duality here: if U ⊂ V is 2-dimensional, then its annihilator U 0 ⊂ V is 4 − 2 = 2-dimensional, so there is a one-to-one correspondence between lines in P (V ) and lines in P (V ).
Algebraische Topologie. Eine Einfuhrung, 2. Auflage by Ralph Stocker, Heiner Zieschang