Witt theorem

Witt 's theorem is a theorem on the properties of finite-dimensional orthogonal spaces over fields of arbitrary type. She argues that any isometry between two subspaces of a finite-dimensional orthogonal vector space can be extended to the whole space.

Formulation

Let be $L$ ${\ displaystyle L}$ - non-degenerate finite-dimensional orthogonal vector space (space with non-degenerate symmetric or skew-symmetric bilinear form ), $L^{\prime },L''\subset L$ ${\ displaystyle L ^ {\ prime}, L '' \ subset L}$ - its two isometric subspaces. Then any isometry $I^{\prime }:L^{\prime }\rightarrow L''$ ${\ displaystyle I ^ {\ prime}: L ^ {\ prime} \ rightarrow L ''}$ can be continued to isometric $I:L\rightarrow L$ ${\ displaystyle I: L \ rightarrow L}$ matching isometry $I^{\prime }$ ${\ displaystyle I ^ {\ prime}}$ on a subspace $L^{\prime }$ ${\ displaystyle L ^ {\ prime}}$ .

Applications

From the theorem of Witt follows the so-called reduction theorem :

Suppose $h$ ${\ displaystyle h}$ non-degenerate quadratic form and shape $h\oplus g_{1}$ ${\ displaystyle h \ oplus g_ {1}}$ equivalent to form $h\oplus g_{2}$ ${\ displaystyle h \ oplus g_ {2}}$ over a field of characteristic not equal to 2. Then the form $g_{1}$ ${\ displaystyle g_ {1}}$ equivalent to form $g_{2}$ ${\ displaystyle g_ {2}}$ over this field.

Literature

Conway J. Quadratic forms, given to us in sensations . - M .: MTSNMO, 2008. - 144 p. - 1000 copies - ISBN 978-5-94057-268-8 .
A.I. Kostrikin , Yu.I. Manin Linear algebra and geometry. - St. Petersburg: Lan, 2008. - P. 304. - ISBN 978-5-8114-0612-8 .

Witt theorem

Formulation

Applications

Literature

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