For any n > 1, 0 < ϵ < 1/2, and N > n ^C for some constant C > 0, we show the existence of an N-point subset X of ℓ ₂ ⁿ such that any linear map from X to ℓ ₂ ^m with distortion at most 1 + ϵ must have m = Ω (min{n, ϵ ^-2 lgN}). This improves a lower bound of Alon [Alon, Discre. Mathem., 1999], in the linear setting, by a lg(1/ϵ) factor. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma [Johnson and Lindenstrauss, Contem. Mathem., 1984].