General Bilinear Forms
Let V be a vector space over a field F equipped with a bilinear form B. We define u to be left-orthogonal to v, and v to be right-orthogonal to u, when B(u,v) = 0. For a subset W of V we define the left orthogonal complement W⊥ to be
There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where B(u,v) = 0 implies B(v,u) = 0 for all u and v in V, the left and right complements coincide. This will be the case if B is a symmetric or skew-symmetric bilinear form.
The definition extends to a bilinear form on a free module over a commutative ring, and to a sesquilinear form extended to include any free module over a commutative ring with conjugation.
Read more about this topic: Orthogonal Complement
Famous quotes containing the words general and/or forms:
“In the drawing room [of the Queen’s palace] hung a Venus and Cupid by Michaelangelo, in which, instead of a bit of drapery, the painter has placed Cupid’s foot between Venus’s thighs. Queen Caroline asked General Guise, an old connoisseur, if it was not a very fine piece? He replied “Madam, the painter was a fool, for he has placed the foot where the hand should be.””
—Horace Walpole (1717–1797)
“There are these sudden mobs of men,
These sudden clouds of faces and arms,
An immense suppression, freed,
These voices crying without knowing for what,
Except to be happy, without knowing how,
Imposing forms they cannot describe,
Requiring order beyond their speech.”
—Wallace Stevens (1879–1955)