The representation dimension is an invariant introduced by Auslander to measure how far a representation infinite algebra is from being representation finite. In 2005, Rouquier gave the first examples of algebras of representation dimension greater than three. Here we give the first general method for establishing lower bounds for the representation dimension of given algebras or families of algebras. The classes of algebras for which we explicitly apply this method include (but do not restrict to) most of the previous examples of algebras of large representation dimension, for some of which the lower bound is improved to the correct value