Question

Problem 6 A bilinear pairing on R2 is given on basis vectors by <ei, ei >= 13; <ei, e2 >=< e2, ej >= 7; <e2,e2 >= 26 a) [3 pts) Find the matrix representation of the pairing. b) (4 pts) Explain why the bilinear pairing defines an inner product. c) [3 pts) If v = [5 – 3]T, find a non-zero vector w with < v, w >= 0

Answer 1

belineat peiring 9 Problem 6: A bilinear on 1R is given vectors by Xe, geq) = 13; Legea>=< €299,7=7 Legg >=26 a) First we define bilinear form onik as <U, U = UTAVE where matrix A EM₂(IR). If @ = [10] and @= [01], then <e,ge, 7= [1,0] [a [a b b][6]=13 • 13 Q=13 20,9933 = [1,07[a 1 2]=1 b=7 le,97= Co, La ][!]=7 c=7 LeyCy> = [0 11 [01101-26 > d=426 Hence, A = [2 26

3 b) Definition C Inner product) det vbe a vector Space over IR. An inner product & ,> is a function VXV IR with 1. tu EV , CU, 4870 and (uu) = 0 4 4 =0 2) &U, VEVgRugv>= {V, 4) 3) & U, V, W EU, and ra, belk holds Lautbr, wy=aRu, wy+b<V, wy det u =(x, y, JT and v=[^2 G₂]T, then <u, u7 = UT AU = [x, y,] y] [92] [91] = [184,774, 73,taky » [y] 130,2 +7 +7 x 9 + 784, +2642 132?+ 142,3,426472 20. (: (x-41² 70 = xh-day 70 and <a,H720, for Xigg, = 0 5°a?y*y; 2xy) 2) Li u =(133 +7 3.74,7269 0 [97] [1923 +74 % + cu 13 7 26 B2 2, +7%, 42 + 7 4 22 +26404 Hence, LU, U) = {v, US 8) Laurby w? = ax + bra 12, tba ay 17 26] Lys = [13Cami+b%) +7(ayıt bys) 7(0%,+68) +26 (ay't by o) ] % ay by:3 [1] Lyz

= 1390, 93 + 13 b X M₂ +79y, dz + Hlay, X3 + 70214877b4243 +2697,93726by: 43 a(139x, +7y, J&3 + (78, +264 D43] + b[C132, + 7b4)23+ (722 +2679343] - a Lu, wy+bLV, WY belinear pairing is inner produc c) ff V = [5-3] , then Hence, the given <V, Wy=[5-3] [13 O = [65-21 35-78 =[44-433131 44x=43 x=434 - a= 43 44 y put y = 44 , yield-n=43 Hence, w= [x 44] T. y] =[ 43