  In this paper, we investigate two families of fully coupled linear
Forward-Backward Stochastic Differential Equations (FBSDE). Within these
families, one could get the same well-posedness of FBSDEs with totally
different structures. The first family of FBSDEs are proved to be equivalent
with respect to the Unified Approach. Thus one could get the well-posedness of
the whole family if one member exists a unique solution. Another equivalent
family of FBSDEs are investigated by introducing a linear transformation
method. By reason of the fully coupling structure between the forward and
backward equations, it leads to a highly interdependence in solutions. We are
able to lower the coupling of FBSDEs, by virtue of the idea of transformation,
without losing the well-posedness. Moreover, owing to the non-degeneracy of the
transformation matrix, the solution to original FBSDE is totally determined by
solutions of FBSDE after transformation. In addition, an example of optimal
Linear Quadratic (LQ) problem is presented to illustrate.
