In mathematics, a bilinear form on a vector space V is a bilinear mapping V × V → F, where F is the field of scalars. That is, a bilinear form is a function B: V × V → F which is linear in each argument separately:
- B(u + v, w) = B(u, w) + B(v, w)
- B(u, v + w) = B(u, v) + B(u, w)
- B(λu, v) = B(u, λv) = λB(u, v)
The definition of a bilinear form can easily be extended to include modules over a commutative ring, with linear maps replaced by module homomorphisms. When F is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument.
Read more about Bilinear Form: Coordinate Representation, Maps To The Dual Space, Symmetric, Skew-symmetric and Alternating Forms, Reflexivity and Orthogonality, Different Spaces, Relation To Tensor Products, On Normed Vector Spaces
Famous quotes containing the word form:
““Now what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of service to them.””
—Charles Dickens (1812–1870)