Show that the functions fx x2 and gx 1x are linearly indep

Show that the functions f(x) = x^2 and g(x) = 1/x are linearly independent on the interval (0, infinity). Show that the vectors V_1 = [-1 2 1 3], V_2 = [1 3 4 7]; and v_3 = [1 1 2 3] dependent by finding a linear dependence relation that they satisfy.

Solution

Let, v3=av1+bv2

So,

1=-a+b

1=2a+3b

2=4a+b

3=3a+7b

a=-2/5,b=3/5

So,

v3=-2v1/5+3v2/5