ISSN:
1420-8997
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, $$\dot V: = \{ z \in V|q(z) \ne 0\} $$ is the set of non-singular vectors ofV, and forx, y ∈ $$\dot V$$ , ∢(x, y) ≔f(x, y) 2/(q(x) · q(y)) is theq-measure of (x, y), where ∢(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping $$\sigma :\dot V \to \dot V$$ we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N∖{0, 1, 2} ∧ ∣F∣ 〉 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F∖{0} such thatF x σ =F x ξ ∀x ∈ $$\dot V$$ andf(x ξ,y ξ)=λ · (f(x, y))ρ ∀x, y ∈V. Moreover, (II) implies ρ =id F ∧q(x ξ) = λ ·q(x) ∀x ∈ $$\dot V$$ , and (III) implies ρ=id F ∧ λ ∈ {1,−1} ∧x σ ∈ {x ξ, −x ξ} ∀x ∈ $$\dot V$$ . Other results obtained in this paper include the cases dimV = 2 resp. dimV ∉N resp. ∣F∣ = 3.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01258504
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