Geometric algebra MOC

Quadratic space

A quadratic space over is a vector space over equipped with a quadratic form , or equivalently*¹ a ^symmetric bilinear form geoalg

The value of is called the quadrance² of .

Further terminology

Let denote the polar form of .

• A vector is isotropic iff , otherwise it is anisotropic; is isotropic iff it has an isotropic vector.
• Iff every vector is isotropic then is totally isotropic.
• A vector is degenerate iff for all , otherwise it is nondegenerate; is degenerate iff it has a degenerate vector and nondegenerate otherwise.
• The set of all degenerate vectors in is called the radical.
• An isometry is a linear map such that for all .
• A bijective isometry is called an orthogonal transformation, and these form the Orthogonal group of a quadratic space

Properties

• Canonical tensors over a nondegenerate quadratic space

See also

• Category of quadratic spaces
• Normal quadratic subspace

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Footnotes

1. Away from 2, see Correspondence between quadratic forms and symmetric bilinear forms away from 2 ↩

2. This term is due to N. Wildberger, which is not to say that I am a wildbergerian. I just like the word. ↩