So that T has rank n and is an invertible matrix.

This would depend on the "characteristic equation" of T, [tex]|T- \lambda I|= 0[/itex] Since every linear transformation satisfies its own characteristic equation, that gives an degree polynomial for T and that can be solved for . That's not quite what you are asking but it's the best I can think of.Then clearly can be written as a linear combination of the basis elements, but can we find this linear combination explicitly? I suspect not...