Summary: We consider the iterative method LSQR for solving \(\min_x |Ax-b|_2\). LSQR is based on the Golub-Kahan bidiagonalization process and at every step produces an iterate that minimizes the norm of the residual vector over a Krylov subspace \(\mathcal{K}_k\). The 2-norm of the error is known to decrease monotonically, although it is not minimized over \(\mathcal{K}_k\). Given a lower bound on the smallest singular value of \(A\), we show that in exact arithmetic the solution lies in the interior of a certain ellipsoid and that the LSQR iterate lies on the boundary of this ellipsoid. We use this result to derive new 2-norm error bounds for LSQR. Although our bounds are not much smaller than the existing ones, we show that they are sharp in the following sense: if the only information we use is our lower bound on \(\sigma_{\min}(A)\) plus the information gained by running \(k\) steps of LSQR, then our bounds cannot be improved. We also show how to choose a point with an error bound smaller than our corresponding bound for the LSQR error, although its true error is not necessarily smaller than the true LSQR error. As LSQR is formally equivalent to the conjugate gradient (CG) method applied to the normal equations \(A^TAx = A^Tb\), we derive analogous error bounds for CG. Our bounds for CG apply to any system \(Ax=b\) where \(A\) is symmetric positive definite.