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 communist society, where nobody has one exclusive sphere of activity but each can become accomplished in any branch he wishes, society regulates the general production and thus makes it possible for me to do one thing today and another tomorrow, to hunt in the morning, fish in the afternoon, rear cattle in the evening, criticize after dinner, just as I have a mind, without ever becoming hunter, fisherman, shepherd or critic.”
—Karl Marx (18181883)
“Your letter is come; it came indeed twelve lines ago, but I
could not stop to acknowledge it before, & I am glad it did not
arrive till I had completed my first sentence, because the
sentence had been made since yesterday, & I think forms a very
good beginning.”
—Jane Austen (17751817)