Just a little nickpick, reading the first lines, I would prefer the author to explain the Pythagorean theorem in R^n like this (is a very simple proof) v . v =

Just a little nickpick, reading the first lines, I would prefer the author to explain the Pythagorean theorem in R^n like this (is a very simple proof) v . v = |v|^2, and u and v are orthogonal (u . v = 0) then |u+v|^2 = (u+v) . (u+v) = u.u + v.v = |u|^2 + |v|^2 A nice example of a projection, if v=(1,1,1,...,1) then the projection of u over v is mean(u)*v