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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.

  3. 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 ...

  4. Classification of Clifford algebras - Wikipedia

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

    The pair of integers (p, q) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted R p,q. The Clifford algebra on R p,q is denoted Cl p,q (R). A standard orthonormal basis {e i} for R p,q consists of n = p + q mutually orthogonal vectors, p of which have norm +1 and q of which have norm −1.

  5. Quadric (algebraic geometry) - Wikipedia

    en.wikipedia.org/wiki/Quadric_(algebraic_geometry)

    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.)

  6. Geometric algebra - Wikipedia

    en.wikipedia.org/wiki/Geometric_algebra

    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.

  7. 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 .

  8. Second fundamental form - Wikipedia

    en.wikipedia.org/wiki/Second_fundamental_form

    and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form L d x 2 + 2 M d x d y + N d y 2 . {\displaystyle L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.} For a smooth point P on S , one can choose the coordinate system so that the plane z = 0 is tangent to S at P , and define the second fundamental form in the same way.

  9. Quadratic algebra - Wikipedia

    en.wikipedia.org/wiki/Quadratic_algebra

    In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups.

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