Question

Determine whether the operator afˆ (x) = f ∗ (x) is linear or nonlinear, where f ∗ (x) is the complex conjugate of f(x).

a. Linear
b. Nonlinear
c. Both
d. Not applicable

Answer 1

The operator afˆ(x) = f ∗(x) is linear because it satisfies the properties of additivity and homogeneity.

Step-by-step explanation:

The operator afˆ(x) = f ∗(x) is linear.

To determine linearity, we need to check if two properties hold: additivity and homogeneity.

Additivity: Let's say we have two functions f1(x) and f2(x). The operator afˆ(x) = f ∗(x) will satisfy additivity if afˆ(f1(x) + f2(x)) = afˆ(f1(x)) + afˆ(f2(x)). Since f ∗(x) is the complex conjugate, we have (f1 + f2)∗(x) = f1∗(x) + f2∗(x), which satisfies additivity.

Homogeneity: Homogeneity means that the operator afˆ(x) = f ∗(x) will satisfy afˆ(kf(x)) = k afˆ(f(x)), where k is a constant. Since the complex conjugate simply takes the conjugate of the function, we have (kf)∗(x) = kf∗(x), which satisfies homogeneity. Therefore, the operator afˆ(x) = f ∗(x) is linear.