Construction of a bilinear symmetric form in orthogonal vector space

Eugeniusz Kusak

Warsaw University, Bialystok

Wojciech Leonczuk

Warsaw University, Bialystok

Michal Muzalewski

Warsaw University, Bialystok

Summary.

In this text we present unpublished results by Eu\-ge\-niusz Ku\-sak and Wojciech Leo\'nczuk. They contain an axiomatic description of the class of all spaces $\langle V$; $\perp_\xi \rangle$, where $V$ is a vector space over a field F, $\xi: V \times V \to F$ is a bilinear symmetric form i.e. $\xi(x,y) = \xi(y,x)$ and $x \perp_\xi y$ iff $\xi(x,y) = 0$ for $x$, $y \in V$. They also contain an effective construction of bilinear symmetric form $\xi$ for given orthogonal space $\langle V$; $\perp \rangle$ such that $\perp = \perp_\xi$. The basic tool used in this method is the notion of orthogonal projection J$(a,b,x)$ for $a,b,x \in V$. We should stress the fact that axioms of orthogonal and symplectic spaces differ only by one axiom, namely: $x\perp y+\varepsilon z \>\&\> y\perp z+\varepsilon x \Rightarrow z\perp x+\varepsilon y.$ For $\varepsilon=-1$ we get the axiom on three perpendiculars characterizing orthogonal geometry. For $\varepsilon=+1$ we get the axiom characterizing symplectic geometry - see [5].

Supported by RPBP.III-24.C6.

MML Identifier: ORTSP_1

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Bibliography

[1] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[2] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[3] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.

[4] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Abelian groups, fields and vector spaces. Journal of Formalized Mathematics, 1, 1989.

[5] Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski. Construction of a bilinear antisymmetric form in symplectic vector space. Journal of Formalized Mathematics, 1, 1989.

[6] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[7] Wojciech A. Trybulec. Vectors in real linear space. Journal of Formalized Mathematics, 1, 1989.

[8] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[9] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.