Let Z be an n × (n − q) matrix of rank n − q whose columns span the space orthogonal to the space spanned by the columns of X and let f = Z′y. Then, on premultiplying by the n × n nonsingular matrix [Z : X], we find that the conventional least squares problem:

may be expressed as

or, equivalently, as

and

where equation (4) defines the optimal value of b in terms of the optimal value of e.

Further, let u be an n × 1 matrix satisfying X′u = c and let d = (Z′Z)⁻¹Z′(a − u) be an (n − q) × 1 function of a. Then the conventional minimum variance unbiased linear estimation problem

may be expressed as

and, on premultiplying by the inverse of the n × n nonsingular matrix [Z : X], we have

Clearly, problems (1) and (3) are algebraically identical to problems (7) and (5) when X, y, b, f, and e are replaced by Z, u, d, c, and a, respectively.