 Rank-deficient ordinary least squares

If a design matrix $X$ has deficient rank, every ordinary least squares minimizer has the form $X^{+}Y+z$ with $z\in \ker X$, where $X^{+}$ is the Moore-Penrose inverse. Hence the fitted values are unique but the coefficients are not: the minimizers form an affine subspace of dimension $p-\mathrm{rank}X$.