I Background in Linear Algebra

In this section we state some elementary results that we will use for our main proofs. I.1 Johnson-Lindenstrauss and subspace embeddings A useful definition for our proofs is the JL moment property, which bounds the moments of the length of Sx. We mention a corollary from [40] which states that JLTs also preserve pairwise angles, which is an important by-product that we will use in our proofs. The next Lemma is part of the proof of [44, Lemma 4.2], which we state here as a separate result to save some space from the longer proofs that follow later. Lemma 4. Let S be a (ϵ, δ)-OSE for a d k matrix U This is part of the proof of [44, Lemma 4.2].