If V is a vector space with an inner space <.,.>. V1 is an non empty subset of V. Vector x is contained in V is said to be orthogonal to v1 if <x,y>=0 for all y contained in V1. 1) if v is contained in V and define the mapping f(x)=<x,v>v. Show f is a linear transformation and describe its range and kernel. 2) if v1 is a subspace of V show that V1 and direct sum of V1 (orthogonal) = V. I tried to prove these as they were stated in a website. But failed. PLease kindly assist me on this matter. Thank.