Condition numbers play an important role in numerical analysis. Classical condition numbers are normwise: they measure both input perturbations and output errors with norms. To take into account the relative scaling of data components or a possible sparseness, componentwise condition numbers have been increasingly considered. In this paper, we give explicit expressions for the mixed and componentwise condition numbers for the weighted Moore--Penrose inverse of a matrix $A$, as well as for the solution and residue of a weighted linear least squares problem $\|W^{\frac12}(Ax-b)\|_2=\min_{v\in\Bbb R^n}\|W^{\frac12}(Av-b)\|_2$, where the matrix $A$ with full column rank.