In this paper, we investigate the normwise, mixed and componentwise condition numbers of the least squares problem min X∈Rn×d ‖AX − B‖F, where A ∈ Rm×n is a rank-deficient matrix and B ∈ Rm×d. The closed formulas or upper bounds for these condition numbers are presented, which extend the earlier work for the least squares problem with single right-hand side (i.e. B ≡ b is an m-vector) of several authors Numerical experiments are given to confirm our results