Additivity and Homogeneity (contd):

To say a system is linear is equivalent to saying the system obeys both additivity and homogeneity.

a) We shall first prove homogeneity and additivity imply linearity.

b) To prove linearity implies homogeneity and additivity.

This is easy; put both constants equal to 1 in the definition to get additivity; one of them to 0 to get homogeneity.

Additivity and homogeneity are independent properties.

We can prove this by finding examples of systems which are additive but not homogeneous, and vice versa.

Again, y(t) is the response of the system to the input x(t).

Example of a system which is additive but not homogeneous:

[ It is homogeneous for real constants but not complex ones - consider ]

Example of a system which is homogeneous but not additive:

[From this example can you generalize to a class of such systems?]