Problem: Given the QR-decomposition of a rectangular matrix $A \in \mathbb{R}^{m \times n}$, where $m > n > 1$, find the QR-decomposition of a matrix $A_k \in \mathbb{R}^{m \times (n - 1)}$, which is matrix $A$ with its $k$-th column removed. The procedure may use $\mathcal{O}(m \cdot (n - k))$ elementary operations ($+$, $-$, $\cdot$, $/$, $\sqrt{}$ etc.)
I think that this could easily be done with Givens rotations, but that would use too many operations because you would have to calculate the new matrix $Q^{(1)}$ by using all those Givens rotations on the old matrix $Q$. The last part of the exercise left me stumped and I have no idea how to continue. This exercise is from an old exam, so it's not homework or whatever. Any help would be greatly appreciated.