If a system is linear, this means that when an input to a given system is scaled
by a value, the output of the system is scaled by the same amount.

In Figure 1(a) above, an input xx to the
linear system LL gives the output yy. If xx is scaled by a value αα
and passed through this same system, as in Figure 1(b), the output
will also be scaled by αα.

A linear system also obeys the principle of superposition. This
means that if two inputs are added together and passed through a linear system,
the output will be the sum of the individual inputs' outputs.

That is, if Figure 2 is true, then
Figure 3 is also true for a linear system. The scaling property mentioned
above still holds in conjunction with the superposition principle. Therefore,
if the inputs x and y are scaled by factors α and β, respectively, then the
sum of these scaled inputs will give the sum of the individual scaled outputs:

Consider the system H1H1 in which

H
1
(
f
(
t
)
)
=
t
f
(
t
)
H
1
(
f
(
t
)
)
=
t
f
(
t
)

(1)for all signals ff.
Given any two signals f,gf,g and scalars a,ba,b

H
1
(
a
f
(
t
)
+
b
g
(
t
)
)
)
=
t
(
a
f
(
t
)
+
b
g
(
t
)
)
=
a
t
f
(
t
)
+
b
t
g
(
t
)
=
a
H
1
(
f
(
t
)
)
+
b
H
1
(
g
(
t
)
)
H
1
(
a
f
(
t
)
+
b
g
(
t
)
)
)
=
t
(
a
f
(
t
)
+
b
g
(
t
)
)
=
a
t
f
(
t
)
+
b
t
g
(
t
)
=
a
H
1
(
f
(
t
)
)
+
b
H
1
(
g
(
t
)
)

(2)for all real tt. Thus, H1H1 is a linear system.

Consider the system H2H2 in which

H
2
(
f
(
t
)
)
=
(
f
(
t
)
)
2
H
2
(
f
(
t
)
)
=
(
f
(
t
)
)
2

(3)for all signals ff.
Because

H
2
(
2
t
)
=
4
t
2
≠
2
t
2
=
2
H
2
(
t
)
H
2
(
2
t
)
=
4
t
2
≠
2
t
2
=
2
H
2
(
t
)

(4)for nonzero tt, H2H2 is not a linear system.

Comments:"My introduction to signal processing course at Rice University."