I'm trying to look for a concise proof for the inequality ||M.v|| ≤ ||M||.||v|| where v is a vector in R^n and M is a matrix with dimensions m*n.
I get the intuition behind what the inequality is saying I think. M is a transformation and it essentially represents how much it can "change" a vector. So if the norm of M is large, it can "change" v by a large amount.
So this inequality is saying that this new matrix Mv cannot "change" vectors by more than a factor of the size of M (||M||).
But my understanding is very abstract and I have no idea how to go about the proof for it. I'm pretty sure the Cauchy Schwartz comes in here somewhere.