Bilinear form

Bilinear form

The geometric description of linear codes

Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016

This means that a k-dimensional bilinear form is described by choosing k2 coefficients aij. The bilinear form can then also be written in matrix notation as follows: define the matrix A = (aij) whose entries are the chosen coefficients. Then (x,y)=∑i,j=0kaijxiyj=xAyt.

Fundamentals of Vector Spaces

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Published in Sandeep Kumar, Ashish Pathak, Debashis Khan, Mathematical Theory of Subdivision, 2019

Sandeep Kumar, Ashish Pathak, Debashis Khan

An inner product space on a real vector space V (V is a vector space over ℝ) is a bilinear form on V×V that associates with each pair of vectors u, v ∈ V, a scalar, denoted by (u,v) or 〈u, v〉 that satisfies the following axioms 〈u,u〉≥0 (Positivity)and 〈u,u〉=0 iff u=0〈u,v〉=〈v,u〉 (Symmetry)(a) 〈u1+u2,v〉=〈u1,v〉+〈u2,v〉 (Additivity) (b) 〈αu,v〉=α〈u,v〉∀u,v∈V and α∈ℝ (Homogeneous)

Attractors of Ginzburg–Landau equations with oscillating terms in porous media: homogenization procedure

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Journal Information

Published in Applicable Analysis, 2023

Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Abylaikhan A. Tolemis

Suppose that is a bilinear form on and let and ( or ). Then the bilinear form defines an inner product on which is equivalent to the inner product

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