Question about linear transformations? So suppose a linear transformation U maps a vector space into itself. Is the rank of U necessarily equal to the rank of its transpose? I know the transpose maps the dual space the same dual space, but I'm not sure how to compute the rank in that case.

At the first you must be show that The annihilator of the range of $U$ is the nullspace of its transpose, then you can conclude the range of $U$ is the annihilator of the nullspace of its transpose. Now we now that the dim of $U$ is equal with dim of its dual, and with the above facts and so that the rank of nullspace addition to rank of range of a linear map is equal to $dim (U)$, similarly dim dual of $U$, you can get that the rank of $U$ is equal to rank of its transpose. 
for more details, you can see the following book in chap 3 and page 27.
Linear Algebra and Its Applications, Peter Lax     
