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:
“The general review of the past tends to satisfy me with my political life. No man, I suppose, ever came up to his ideal. The first half [of] my political life was first to resist the increase of slavery and secondly to destroy it.... The second half of my political life has been to rebuild, and to get rid of the despotic and corrupting tendencies and the animosities of the war, and other legacies of slavery.”
—Rutherford Birchard Hayes (1822–1893)
“The most passionate, consistent, extreme and implacable enemy of the Enlightenment and ... all forms of rationalism ... was Johann Georg Hamann. His influence, direct and indirect, upon the romantic revolt against universalism and scientific method ... was considerable and perhaps crucial.”
—Isaiah Berlin (b. 1909)