Let V be inner product space with dim(V)<inf.
Let T:V-->V be a linear transformation. Prove:
1. (ImT)^# = kerT*
2. (kerT)^# = imT*
(*=unitary operator )
(#=orthogonal complement )
Your last three questions are standard stuff. You can find it in any decent linear algebra book. Check, for example, the book "Algebra" by Prof. S. Amitzur, or the one by Prof. Levi, both in hebrew, Ed. Akademon , The Hebrew University.