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
Famous quotes containing the words matrix and/or form:
““The matrix is God?”
“In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
“If it has limits, it isn’t omnipotent.”
“Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
—William Gibson (b. 1948)
“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 any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!”
—Charles Dickens (1812–1870)