  In this paper, based on the theory of adjoint operators and dual norms, we
define condition numbers for a linear solution function of the weighted linear
least squares problem. The explicit expressions of the normwise and
componentwise condition numbers derived in this paper can be computed at low
cost when the dimension of the linear function is low due to dual operator
theory. Moreover, we use the augmented system to perform a componentwise
perturbation analysis of the solution and residual of the weighted linear least
squares problems. We also propose two efficient condition number estimators.
Our numerical experiments demonstrate that our condition numbers give accurate
perturbation bounds and can reveal the conditioning of individual components of
the solution. Our condition number estimators are accurate as well as
efficient.
