Notice that if z_1 and z_2 are both solutions of z' * x = 1, then z_1 - z_2 is a solution to the equation z' * x = 0. What can you say about the set of solutions z to the equation z' * x = 0?
________________________________ Eric J. Wingler (firstname.lastname@example.org) Dept. of Mathematics and Statistics Youngstown State University One University Plaza Youngstown, OH 44555-0001 330-941-1817
"junoexpress" <email@example.com> wrote in message news:firstname.lastname@example.org... > Hi, > > I am curious if there is a general way to understand the solution to > the following (simple) complex analysis problem. > > Suppose we have an n-dimensional vector space, and a fixed (i.e. known) > vector x in C^n. > > The question is whether there is a way to describe the set of all > vectors z such that: > (i) z' * x = 1 > (where ' denotes conjugate transpose). > > If you sketch out this problem, it is not difficult to see that the > conditions: > (ii.a) Re(z' * x) = 1 > and > (ii.b) Im(z' * x) = 0 > give you two linear equations in 2n unknowns, which you could then > solve (in a least squares sense). > > This method of analysis, however, does little to describe what the > solution set is like. I am curious if anyone else has another way of > thinking about this problem. More general pictures for how to visualize > this condition (like as a projection for example) do not seem that easy > to conjur up. > > Thank you for any help you can provide, > > Juno >