Problem 3 (10 points). Gram-Schmidt process. Consider a basis {v1, ..., vx} of a k-dimensional subspace W of a vector space V equipped with an inner product (, .). Let u1, .. ., ux be defined as U1 = v1 U2 = 12 - proju, (v2) ug = V3 - proju, (v3) - projuz (v3) k- 1 up = Uk proju; (Uk), j=1 where the projection operator proj is defined as proj, (v) = ju for every u, v E V with u * 0. Lastly, let ui = Juill for i e {1, ...,k}. Prove that {u1, ..., uk} is an orthormal basis of W, i.e. {ul, . .., uk} is a basis of W and is such that (un, uj) = bij