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Using homogeneous coordinates, a non-zero quadratic form in n variables defines an (n − 2)-dimensional quadric in the (n − 1)-dimensional projective space. This is a basic construction in projective geometry. In this way one may visualize 3-dimensional real quadratic forms as conic sections.
The definition of a projective quadric in a real projective space (see above) can be formally adapted by defining a projective quadric in an n-dimensional projective space over a field. In order to omit dealing with coordinates, a projective quadric is usually defined by starting with a quadratic form on a vector space. [4]
By definition, a quadric X of dimension n over a field k is the subspace of + defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables , …, +. (A homogeneous polynomial is also called a form, and so q may be called a quadratic form.)
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system (,,) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces , the ellipsoidal coordinate system is based on confocal quadrics .
bicomplex numbers: a 4-dimensional vector space over the reals, 2-dimensional over the complex numbers, isomorphic to tessarines. multicomplex numbers: 2 n-dimensional vector spaces over the reals, 2 n−1-dimensional over the complex numbers; composition algebra: algebra with a quadratic form that composes with the product
Given a finite-dimensional vector space over a field with a symmetric bilinear form (the inner product, [b] e.g., the Euclidean or Lorentzian metric) : , the geometric algebra of the quadratic space (,) is the Clifford algebra (,) , an element of which is called a multivector.
A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form.
Essential dimension of quadratic forms: For a natural number n consider the functor Q n : Fields /k → Set taking a field extension K/k to the set of isomorphism classes of non-degenerate n-dimensional quadratic forms over K and taking a morphism L/k → K/k (given by the inclusion of L in K) to the map sending the isomorphism class of a quadratic form q : V → L to the isomorphism class of ...