Question

Determine whether the given set of functions is linearly independent on the interval (−∞, ∞)

f1(x) = x

f2(x) = sin(x)

f3(x) = sin(2x)

Answer 1

given that = sina f, 1x) = x falx) w) x 6 - 4 , f3/2) sim 2x that soi? هوا know lineonly & fz independent fi are fa iff 0 x x wlf, fz, f3) (x) wronskian of f, (2), fz (x) & f3 (2) is defined f, 12) f(x) fz/v) ay tila 11 wife, frita) 1x) = f2 (x) 15" 3) f" (2) sina simar x 11 cox x a cosax no sina - usimax = xl-4c38x sinex + 2 cos zx sine) -6 u sinax sima + simax sine) .

w (for fre to)(x) = x (-4 COS X Sim2 2 + 2 608 24 sima) + 3 simax sima bul x = 0 wlfo, fri fa) (0) (0) = 0 iie x = 0 are not wl for fac f2) (x) = 0 at => f, (2), fz/x) & fəlx) linearly independent =) 6,16), fe/z) 4 f/4) are linearly dependent