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By Herbert Möller

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Now assume r > 1 with the appropriate induction hypothesis. Define Ur_1 = Ful + + Fur-1- So Ur_1 C U and hence Ur 1 D U-L. Pick y E Ur i - U-L, and let Hr = FUr + Fy. -1 C Hr Now Ur_ i is an (r - 1)-dimensional totally isotropic subspace of the regular space Hr , so by the induction hypothesis Hr has . an orthogonal component of the form Hi for 1 < i < r - 1. The result follows. 5. 44 (Witt's Isometry Extension). : U -p V' is an isometry into V'. Then a. extends to an isometry V -p V'. Proof. First check that the result is easy to achieve if U is regular.

1). We will see later that the numbers of positive and negative terms in this expression are invariants of V. (iii) Suppose F is a finite field of odd order q. The group homomorphism F* --* F* with action a --* a2 has kernel {+1}. It follows that (F* : F*2) = 2. Therefore there is a splitting V ^J (1, . 1, E) ... , e), where E is a fixed . 3. Representation, Splitting, and Isotropy; Invariants u(F) and s(F) 23 nonsquare in F. We will see in Chapter 3 that the basis can be chosen so that c appears at most once in this expression.

39. In the preceding proof, having obtained the congruence Z 1 U1 Z' 1 W1, with Z Z', the reader may have felt a temptation to invoke the induction hypothesis and thereby cancel Z and Z' to get the conclusion. The fly in this ointment lies in the fact that the induction hypothesis deals with cancellation of two isometric subspaces of the same space, whereas there is no reason to believe that the spaces Z 1 U1 and Z' 1 W -L are the same. Once the isometry A has been applied, however, this objection is overcome, so the cancellation (based on the induction hypothesis) becomes acceptable.

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Algorithmic Linear Algebra by Herbert Möller

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