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  2. Quadratic form - Wikipedia

    en.wikipedia.org/wiki/Quadratic_form

    A mapping q : M → R : v ↦ b(v, v) is the associated quadratic form of b, and B : M × M → R : (u, v) ↦ q(u + v) − q(u) − q(v) is the polar form of q. A quadratic form q : M → R may be characterized in the following equivalent ways: There exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form. q ...

  3. Essential dimension - Wikipedia

    en.wikipedia.org/wiki/Essential_dimension

    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 ...

  4. Clifford algebra - Wikipedia

    en.wikipedia.org/wiki/Clifford_algebra

    A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...

  5. Classification of Clifford algebras - Wikipedia

    en.wikipedia.org/wiki/Classification_of_Clifford...

    This article uses the (+) sign convention for Clifford multiplication so that = for all vectors v in the vector space of generators V, where Q is the quadratic form on the vector space V. We will denote the algebra of n × n matrices with entries in the division algebra K by M n (K) or End(K n).

  6. Definite quadratic form - Wikipedia

    en.wikipedia.org/wiki/Definite_quadratic_form

    In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the

  7. Witt's theorem - Wikipedia

    en.wikipedia.org/wiki/Witt's_theorem

    "Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.. In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the whole space.

  8. Milnor K-theory - Wikipedia

    en.wikipedia.org/wiki/Milnor_K-theory

    Finally, there is a relation between Milnor K-theory and quadratic forms. For a field F of characteristic not 2, define the fundamental ideal I in the Witt ring of quadratic forms over F to be the kernel of the homomorphism () / given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:

  9. Isotropic quadratic form - Wikipedia

    en.wikipedia.org/wiki/Isotropic_quadratic_form

    A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V. Then W is called an isotropic subspace of V if some vector in it is isotropic, a totally isotropic subspace if all vectors in it are isotropic, and a ...

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