Question

Determine whether the following operators are linear or nonlinear.

a) Âf(x)=SQUARE f(x)
b) Âf(x)=f *(x) (Hint: form the complex conjugate of f(x))
c) Âf(x)=0 (multiply f(x) by zero
d) Âf(x)=[f(x)]-1 (take the reciprocal of f(x))
e) Âf(x)=f(0) (evaluate f(x) at x=0)
f) Âf(x)=ln f(x) (take the logarithm of f(x))

Answer 1

Linear operators in mathematics follow the principles of additivity and homogeneity. When analyzing the given operators:

a) Âf(x)=SQUARE f(x) is nonlinear.

b) Âf(x)=f *(x) (using the complex conjugate of f(x)) is nonlinear.

c) Âf(x)=0 is linear.

d) Âf(x)=[f(x)]-1 (taking the reciprocal) is nonlinear.

e) Âf(x)=f(0) is linear.

f) Âf(x)=ln f(x) (taking the logarithm) is nonlinear.

Step-by-step explanation:

Linear operators satisfy two conditions: additivity and homogeneity. Additivity means that the operator of the sum of two functions equals the sum of the operators applied to each function individually. Homogeneity refers to the operator of a scaled function being equal to the scalar multiplied by the operator of the original function.

For instance, option c) Âf(x)=0 is a linear operator since multiplying any function by zero results in the zero function, satisfying both additivity and homogeneity. Option e) Âf(x)=f(0) is also linear because evaluating a function at a specific point (x=0) doesn't violate the conditions of linearity.

On the other hand, options a), b), d), and f) are nonlinear operators. They don't satisfy both additivity and homogeneity simultaneously. For example, option a) Âf(x)=SQUARE f(x) doesn't follow additivity, as the square of the sum of two functions doesn't equal the sum of the squares of the functions individually, violating linearity.