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:
“There is absolutely no evidence—developmental or otherwise—to support separating twins in school as a general policy. . . . The best policy seems to be no policy at all, which means that each year, you and your children need to decide what will work best for you.”
—Pamela Patrick Novotny (20th century)
“How superbly brave is the Englishman in the presence of the awfulest forms of danger & death; & how abject in the presence of any & all forms of hereditary rank.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)