Question

(1 point) Are the functions f, g, and h given below linearly independent? f(x) = 621 + cos(9x), g(x) = 621 – cos(9x), h(x) = cos(9x). If they are independent, enter all zeroes. If they are not linearly independent, find a nontrivial solution to the equation below. Be sure you can justify your answer. (e24 + cos(9x)) + (e21 – cos(9x)) + (cos(9.x)) = 0.

Answer 1

g O Griver functiom arce f(2)=ffet cos (98), 9 (2)=&-cos (90), n(a)= los (92)- Now, Wronkskian (f, g, h) f h ire, w(18 h) = g' h da, g" "+ Cos (92) €29_cos (92) [2922_g sin(92) 26 +9 Sin (92) - Sin (92) Luet - 81 cos(93) 4€29+81 CO (0%) -81 Cos (92) Here ta dff" d? da2 18" e22 los (9) 29 0 At a=0 w(d.gh) 2 2 |-77 85 -81 0 0 - [eſ-4-20 2 2 0 85 85-81 ço [": 1st column zand 'column] ::8,9,6 are linearely dependent. Thus, if we write linear combination of 8 (22.8190 and n (2) to zero on a fra] + ezgrastegh ()0 then & , C2, C3 are not all zeros. :: 4 [629420s (93)]+62 [649_Cos ( 9x]] tez Cos (93)=0 >> (+6) + (-69+) Cos (9x) = 0.19 +0. Cas Cox comparing both sides we get G + C2 20 > la-G --C-e, G-C2+63 2 0 » (3222-9 =-29

) Since, G , l2, 23 are not all zeros. Then let Gak (to) (2=-k, e32-2k N Now, taking kal We get, 421, C2=-1, €33-2 lie, 1.812) - 1.81a) -2.690 0 (84 cas (92) + (-Cos (92) 21 c eos (92)) = 0 +