Abstract: For any n>1 and eps in (0,1/2), we show the existence of a poly(n) sized subset X of R^n such that any linear map from (X, l_2) to l_2^m with distortion 1+eps must have a target dimension satisfying m = Omega( min{ n, eps^{-2} log n } ). The proof is simple, and all details will fit in the talk. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma, and is thus optimal. Previously, Alon proved a bound against all embeddings (not necessarily linear), but which was worse by a log(1/eps) factor. I will close the talk with evidence of why I believe it might be possible to beat the JL lemma using non-linear maps. Joint work with Kasper Green Larsen.