Lecture 14 — Thursday , Oct 17 2013

Proof. For the m = d case, first rotate the subspace W to become span(e1, . . . , ed) (via multiplication by an orthogonal matrix), and then project to the first d coordinates. This clearly preserves norms in W exactly. Now, assume there is an ε-subspace embedding Π ∈ Rm×n for m < d. Then, the map Π : W → Rm has a nontrivial kernel, in particular there is some w ∈ W,w 6= 0 such that Πw = 0. On the other hand, ‖Πw‖2 ≥ (1− ε)‖w‖2 > 0, contradiction.