Question

(1 point) Are the following statements true or false? ? 1. If z is orthogonal to uị and u2 span(uj, u2), then z must be in and if W = Wt. ? 2. For each y and each subspace W, the vector y – projw(y) is orthogonal to W. ? 3. If y is in a subspace W, then the orthogonal projection of y onto W is y itself. ? 4. The orthogonal projection p of y onto a subspace W can sometimes depend on the orthogonal basis for W used to compute p ? ? 5. If the columns of an n x p matrix U are orthonormal, then UUTy is the orthogonal projection of y onto the column space of U

Answer 1

1. If z Ps orthogonal to ug and 42 and if Wespan 14, 42) then a must be in the Ans: - Toule Justification & As <2,4,>=0 z is orthogonal to 4,442, <2,427 0 veltor vis Waspaneli, 42) V= QU4 + 642 are scalar consider <z, v) = <te, au, +6427 = á <2,4,7+ 5 <2,427 for any a, b = 4 (0)+5 (6) = 0 → <2,170 z is oothogonal to for every vel z & Wt and each subspace wl, the rector is orthogonal to w e for each Я y- proje (4) Anso - True Justº fications - Let basis of W. ¢ {Vig V27.,Vn} be an orthonormal di, da, ..., do are scolares Let wel Vad vitda Vat ...tdown <Vn, 47 NO Now <V, 47 V - 212, 47 V2 4 - provides c 4- proginya) consider av, u - projw.243>=29,47-2v, - KV,43 - ev, avi, Y> V7+ <V2,4% 12, forritun, ys unt - <V, > - <w:y><V, V7 - 4124> «v, Va> <vn,y> <v, vnd <V,47 - (V,47 SO

4- pooley is orthogonal to ul N A . ५ 9. If yes on a subspace. W then en the orthogonal projection of y onto Nie itselt Ans: True Justification :- Let {viivag...., Vn} be an orthonormal bar 4 of an YEW y = di vi tog Vata. tan Vn & uits are scolar also #i=1, 2, ..., 1 Now La, v podine <4xVis V,+ 4,127 Il 22/12 Vet.no + ۴۷۱۱۱۹ HVIS : 11 VI) = 1 - 22 Vatt + erano T =livit & Vattan ay The oothogonal projection pofy onto a subepace can sometimes depend on the orthogonal balls for e used to compute p Ansu - False Justification : Oethogonal peojection of poly onto subspace is independent of bethago neel batis

(5) If the columns of an xp manex U are orthio normal then wuty & the orthogonal projectory of y onto the column space of u Answerd True