Question

Problem 2 Determine if the following functions are linearly independent or linearly dependent. If you believe that they are linearly dependent (i.e. W(5,9) (+) = 0, for all t in some interval) find a dependence relation. 1. f(t) = cost, g(t) = sint 2. f(t) = 61, g(t) = 64+2 3. f(t) = 9 cos 2t, g(t) = 2 cos? t - 2 sinat 4. f(t) = 2t>, g(t) = 14

Answer 1

817) = siut fet) = Cost w(f:37 CV) : lost Sut a lost t sint = 1 40 Sint Cast so fety and get are note linearly dependent t12 fex) = 6 Mt) - 6 w (f,g) t = 6r 6t 2 6t6t ） et d 6 st leit 6 6 6 6t 6 st lub- 66t.stens o . so fit) and sit) are linearly defendent. and the Relation of 69. fet)= 827) dependence is

3. fit) = 9 Coret (t) = 210st - 2siunto 2 203 2t (6.09 (1 9 Bit 2003 at Sinzi -4 18 Sint Wor • 36 20324 Siu2t 4.36 Coszt Su 26 c and so, fo and Mt) are linearly defendent 233) Relation is; 2fet) = 99+) (fic) - 9ght 4 f« -, 24", NL) = 41 t9 (1-3) () - 242 کہا 3 4t. 8ts-qt5 = ats to except to so, fit) and (t) are of linearly dependent expcept, too. are learly deperdent fit) at t=o at to They and 9ct) Ffet)=914) ری Relation,