Question

Linear Algebra.

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4. (Total Points=10) Show your work for full credit Let P be the vector space of all polynomials of degree at most one p ao a1 and q = bo + b1ax to be < p, q >= aobo + ajbi. We define < -,- > for a)(5 points) Show that < -, - > is an inner product.

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Answer 1

Po be the vector space of all polynomials of degree at most one! Define asi for pzas para وره ۵ ماده * كه ممط مهدومك ها - | @ ☺ let (n)= ao otain, que botbin, r(~) = cofen. < p (m) + 2(~), 8(~)>= aotantoofbin, cofein) = ((aotbo) + (c4 +61) n, Cofan> = factbo).c. + (Q4 +61).@ dolotbocopa at big = faolotac) + (boco faci) Castan, coeam> t<bot bin, copan> = 5pm) ir(n)) (4m), (m) W et aff. p(u) = aofan q(n) = bot labin Tap(on), q (m)>=<xaotarym, bo botoim) - axao by + xy.bi salaobo fubi) =a.saotan, botbin) =&& polon) 1310n]). 6 let (m) = aotain, q (m) = botbin I : <b(m) 9(0m)) = ao botas Got brao .= bodo tb, al = <g(n), blond Tu let pln) = ao faim, qf(x) = boutbol <k(om), 6(n)) = <aofan, nolan) 2 ao aff as a laolt 1912 70. sklon), b[n)>=0 = laul Floritzo o laol s04 1012 = - ao o a=0 = p (m)=0.

& If þ(m) = 0 =0fare on Then sp(m), b (on)) so 7) Tb (m) , p (m2) zo iff blom) zo <,) is an inner product s={ 2+6m, .3-n}. p(om) = 2t6m, q\")=3-m p(m) anya (atom, 3-n) = 2.3 +6.1-1) = 6-6 =0 is is orthogonal, © let u,= blon) = Roos 2+6on u z him) of (a), p(m) ada) T Fhrany's blant = 3-m - © - When 3pcon) ,q(om) } is an orthogo nal set, I pron) - g(n) 3 B an ortho nomal set. 2 Hom) 112(on) . Now 11 p(n)Il a k plne) p (m)) = 2.2 +6.6 = J 46 19 (mill = 59(2) 19 (1) √ 3.3+(1) El agt z Jo. is orthonormal. 40 2+6m :- So } is orthomemad.