The Matrix of A Bilinear Form
A bilinear form on a vector space V over a field R is a mapping V × V → R which is linear in both arguments. That is, B : V × V → R is bilinear if the maps
are linear for each w in V. This definition applies equally well to modules over a commutative ring with linear maps being module homomorphisms.
The Gram matrix G attached to a basis is defined by
If and are the expressions of vectors v, w with respect to this basis, then the bilinear form is given by
The matrix will be symmetric if the bilinear form B is a symmetric bilinear form.
Read more about this topic: Change Of Basis
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