Final answer:
The functions f(x) = e^(5x) + cos(3x), g(x) = e^(5x) - cos(3x), and h(x) = cos(3x) are linearly dependent, not independent.
Step-by-step explanation:
The functions f(x) = e^(5x) + cos(3x), g(x) = e^(5x) - cos(3x), and h(x) = cos(3x) are linearly dependent, not independent.
To check for linear independence, we need to see if there are any constants c1, c2, and c3, not all equal to zero, such that c1*f(x) + c2*g(x) + c3*h(x) = 0.
By substituting the given functions, we have c1*(e^(5x) + cos(3x)) + c2*(e^(5x) - cos(3x)) + c3*(cos(3x)) = 0.
Since e^(5x) and cos(3x) are linearly independent, we can rearrange the equation to (c1 + c2)*(e^(5x)) + (c1 - c2 + c3)*(cos(3x)) = 0.
This implies that both (c1 + c2) and (c1 - c2 + c3) are equal to zero, which results in c1 = c2 = c3 = 0. Therefore, the functions f, g, and h are linearly dependent and not independent.