We prove a finite family developments theorem (FFD) for the calculus with explicit substitutions Linear Substitution Calculus (LSC). The notion of redex family is obtained by adapting Lévy labels to support the two distinctive features of LSC, namely its use of context rules that allow substitutions to act “at a distance” and also the set of equations modulo which it rewrites. We then apply FFD to prove a number of results for LSC including: an optimal reduction result, an algorithm for standardisation by selection, and normalisation of a linear call-by-need reduction strategy.