Question

(1 point) Let {uj, u2, u2 ) be an orthonormal basis for an inner product space V. Suppose y = qui + buz + cuz is so that|lvl1 = V116. (v, uz) = 10, and (v. uz) = 4. Find the possible values for a, b, and c. a = CE

Answer 1

The answer is given as
net (U, 42, 43) be an ertho normal basis Jos on inner product space v. Suppose V = au + buz & Cuz sot lvl1 = VIIG and {1,42) = 10 es LV, 43>= 4 we have to find possible value of a, b, c & (ull U2, U3) be orthonormal basis- since so & 124,1l = 1uzll = 1143)1 = 1 <u,;42) =<U2, U3> = {43, 4,) in Now CX, 42) = 10 <auitbu2 +(U3, 42) = 10 - = a <41,423 + b [42, 42) + C(43, 42) = 10 Tb=10] {from @} Now <V, Ug) = 4 Kaufbuz & C431 43 = 4 akui, 43> +6 LU₂, 43> +C<43143) = 4 [c=4 »

د ۱۱۱۱۱ 116 تنو N - Lau,+b42+Cziga upfb42+ (43>=116 =) Laulau,+buzt (U3) & Lbluz, a utby2fcuz) ) = 116 + <cu, au, + bluz of Cle = a lui, au,+buz & CU3> tb Luz, au, Abu2fcu? & cluz, a ufb4₂4 CU3) = 116 a cui , auista Luin buz>t a Lui, CU> tb<uz, au> tbluz, Du 27 & bluz, cuz) te Luz, aui) & (uz, buzz &ccus, Cugal

= a. a <u, uz 3 ta b Lul, 42) + a.ł <U, U tb. a Luz, 4)+ b. b <42, 422 tb. c < 42, + c. a 143, U) & C.5 43, 4₂) + C. c <U3, U37 = 116 - 11a112 114,112 +ofoto + 116 112 1142) to foto +ucl12 1143 11² = 116 =) 11a12 + 11 bil? + 11611² = 116 =) a² + b² + 2 = 116 - a² + 100 + 16 =116 - a2 + 16 = 116 Taco Men & tuo old to dusjuos v= o.li+ 10 4₂ of 4 uzo USU

Let Vir V2, V3 is an 2 V3 is an ors in IRS span (vi V2, 3) vi v = 42 Uz - Uz = 59 orthogonal set of Let w be a Vector such that in wod=-126 w.dz = 236 w. Uz = 9 o Espon (V., Oz, ws) we & Vit B iz trwy - Now wedi = – 126 IV - (a Vit B dz + V63). Vi = -12 6. av. Vi = -126 & (42) =-126 a = – 126 42 12=-3] un Now w. Oz = 236 (2 V + B V2 + ruz). V2 = 236 PUzUz = 236 -- -

ß (59) = 236 IP = 42 Now (& Q, +B Qz + roz). V = 9 - avi. Uz + B V2 - Vzt roz. 83 = 9 -> r.(Uz. 1x)= 9 V=D hence w=-3 V + 4 Uz toz