dimension theorem for symplectic complement (proof)
We begin by showing that the mapping , is an linear isomorphism. First, linearity is clear, and since is non-degenerate, , so is injective. To show that is surjective, we apply the http://planetmath.org/node/2238rank-nullity theorem to , which yields . We now have and . (The first assertion follows directly from the definition of .) Hence (see this page (http://planetmath.org/VectorSubspace)), and is a surjection. We have shown that is a linear isomorphism.
Let us next define the mapping , . Applying the http://planetmath.org/node/2238rank-nullity theorem to yields
Now and . To see the latter assertion, first note that from the definition of , we have . Since is a linear isomorphism, we also have . Then, since , the result follows from equation 1.
|Title||dimension theorem for symplectic complement (proof)|
|Date of creation||2013-03-22 13:32:52|
|Last modified on||2013-03-22 13:32:52|
|Last modified by||matte (1858)|