- Problem 10 Let Ui U2 Un Denote A Basis Of An Inner Product Vector Space V Con Sider The Set Of Vectors 01 02 1 (56.6 KiB) Viewed 26 times
- Problem 10 Let Ui U2 Un Denote A Basis Of An Inner Product Vector Space V Con Sider The Set Of Vectors 01 02 2 (20.02 KiB) Viewed 26 times
Problem 10 Let {ui, u2,...,Un} denote a basis of an inner product vector space V. Con- sider the set of vectors {01, 02, ..., Un) obtained as - 2 -V- Vj = 11 (U2, U1) U2 = t12 011 lloy (tus, vy) (U3, 22) U3 U- U- 02, || , ! || 2012 | (Un, va) (un, Un-1) V., 02-...- Un-1 || 0111 ||l? llun-1 // Use induction on n (n > 2) to show that the set {U1, U2, ..., Un} is orthogonal by filling in the following. (Throughout the proof you will use a 1b <a,b >=0) BASE STEP (n=2) Explain why or show that the set {01, 02} is orthog. onal, that is Vi = 21 and (u2, 01) 2 U2 01 le 112 are perpendicular. INDUCTION HYPOTHESIS Assume the set {V1, V2, ..., Un-1} is or- thogonal, that is any two distinct vectors in the set are perpendicular, equiva- lently <V, V, >=0 for every i j = 1,2,..., n-1. INDUCTIVE STEP Show that the set {V1, 02,..., Un-1.0.) is orthogonal. Because of the induction hypothesis, the claim reduces to proving that on is 3
perpendicular to vi, 02, ..., Un-1. Thus to prove the inductive step you will need to prove (Ura, ) Vi-...- U-1, >=0 (un, v1) < Un, vị >< tin - 01-...- wall? for every i = 1,2,..., 1-1. |vi| || 0-1 ||