Question

Test the set of functions for linear independence in ℱ. If it is linearly dependent, express one of the functions as a linear combination of the others. (If the set is linearly independent, enter INDEPENDENT. If the set is linearly dependent, enter your answer as an equation using the variables f, g, and h as they relate to the question.) {f(x) = 8, g(x) = sin(x), h(x) = cos(x)}

Answer 1

Linearly independent

Explanation:

let a,b,c be element of F. for all x element of F.

1) if a, b and c are zero then they are independent

2) if not all a,b,c are zero then it is independent.

Now lets write it as a linear combination

i.e 8a +b Sinx +c Cosx = 0

equating the coefficients we have

: 8a =0 hence a = 0

: b Sinx = 0

b = 0

: c Cos x = 0

c is not 0

Hence it is Linearly independent