The problem of bumpless transfer is posed in the linear quadratic context, and formulae derived for a full-information gain which minimises a certain cost function. These formulae incorporate two weighting matrices to add flexibility to the design, and they can also be used when the controllers in question are non-invertible. Results are given for both continuous and discrete lime cases, and both 1 and 2 DOF bumpless transfer is considered. When extended to the infinite horizon, the formulae depend only on known matrices and the solution to a single Riccati equation. Finally, it is shown that the formulae given here reduce to the Hanus Conditioned Transfer formulae under certain circumstances; that is, the results here could be considered as a generalisation of the Hanus technique.