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:
“According to the historian, they escaped as by a miracle all roving bands of Indians, and reached their homes in safety, with their trophies, for which the General Court paid them fifty pounds. The family of Hannah Dustan all assembled alive once more, except the infant whose brains were dashed out against the apple tree, and there have been many who in later time have lived to say that they have eaten of the fruit of that apple tree.”
—Henry David Thoreau (1817–1862)
“All forms of government symbolize an immortal government, common to all dynasties and independent of numbers, perfect where two men exist, perfect where there is only one man.”
—Ralph Waldo Emerson (1803–1882)