Recall that for a subvector space W of complex vector space V, the orthogonal complement is WA {VEV : VWEW (v, w) = 0} In what follows, assume that V is of finite dimension. [Q1] Show that W- is a vector subspace. [Q2] Show that are in direct sum, denoted WOW! = V. (check the definition if you do not remember it!). Also show that dim(W) + dim(WL) = dim(V) (Hint: this should follow as part of your proof in which you'll use an orthobasis). [Q3] Show that (WI) - = W. Now, let H E Rmxo be a real valued matrix. The column space (also called image or range) of H is Im(H) 4 {Hx : X ER" ] CRm The kernel (also called null space) is ker (H) 2 {x : Hx = 0} CIR". The objective of this problem is to show a few properties that are useful in analyzing linear regression. [Q4] Show that ker(H) = Im(HT)+, ker(HT) = Im(H)+, Im(H) = ker(HT)I, Im(HT) = ker(H) . (Hint: don't prove four properties... if you prove one the others follow using the previous questions [Q5] Show that Im (HTH) = Im(HT) = row(H). [Q6] Show that ker(HTH) = ker(H)