Question

Determine whether the given set of functions is linearly independent on the interval (-00,00) fı(x) = xf2(x) = sin(x) $3(x) = sin(2x)

Answer 1

Consider the given functions: ti(n=a. f(n)= senn, f(n)= rin an. for ne (100,00) Suppose C, G, C₂ EIR, and. Cofit Cate + c3f3=0. then cint Ca linn. & Caren 2n=0 On differentiating with respect ton; Cit C₂ corn & 2 C3 Corrn=0 . -0 2 Now again differentiating with respect tox, we get 0 + 6(2) sein + (-2(3). 2 Benanzo, 3

Here if we get the c_{​​​​​​1} = c_{​​​​​​2} = c_{​​​​​​3} =0, then these three functions will be linearly independent otherwise these will be linearly dependent.

If we put na it in the equation Oi cix+0+0=0 -) Ciro Now if we put x= N21 then. equation ) is: of Cz. (or nl2 + 2 (z Casa = 0 C2.0 +26 cs=0 → (3 - 0 Now put n=alg in the equation (3) 1 we have + (C2).1 + (-4) (3 lent=0 - (q +0=0 (2=0 Thus, we get the trivial solution. C = (2=(₂=0 le ere Therefore fo fai to liearly independent