Question

(8)determine whether the given set of function is linearly independenton interval (-[infinity],[infinity])f1(x)=5 f2(x)=cos^2x f3(x)=sin2(x)andf1(x)=cos2x f2(x)=1 f3(x)=cos^2(x)

Answer 1

The set of functions {f1(x) = 5, f2(x) = cos^2(x), f3(x) = sin(2x)} is linearly independent on the interval (-∞, ∞).

The set of functions {f1(x) = cos(2x), f2(x) = 1, f3(x) = cos^2(x)} is linearly dependent on the interval (-∞, ∞). This is because f3(x) can be written as a linear combination of f1(x) and f2(x):

f3(x) = (1/2)f1(x) + (1/2)f2(x) + (1/2)

Therefore, f3(x) is not linearly independent of f1(x) and f2(x).