The function obeys the scaling/homogeneity property, but doesn’t obey the additivity property, thus not being linear.

Explanation:

The linearity of a system can be determined by applying the principle of superposition. If the output of the system is proportional to the input, then the system is linear. Mathematically, this can be expressed as follows:

y[n] = a*x[n] + b*h[n]

where y[n] is the output, x[n] is the input, h[n] is the impulse response of the system, and a and b are constants.

In the given system, y[n] = n*x[n]. Let us apply the principle of superposition to check the linearity of this system.

1. Homogeneity property:

If the input is scaled by a constant factor, then the output should also be scaled by the same factor.

Let us apply a scaling factor of a to the input:

y1[n] = n*a*x[n]

Now, let us apply the same scaling factor to the output:

y2[n] = n*a*x[n] = a*y[n]

Since y1[n] is not equal to y2[n], the system does not satisfy the homogeneity property. Therefore, the system is not linear.

2. Additivity property:

If two inputs are applied simultaneously, then the output should be the sum of the individual outputs.

Let us apply two inputs x1[n] and x2[n]:

y1[n] = n*x1[n]
y2[n] = n*x2[n]

Now, let us apply both inputs simultaneously:

y[n] = y1[n] + y2[n] = n*x1[n] + n*x2[n] = n*(x1[n] + x2[n])

Since y[n] is equal to n*(x1[n] + x2[n]), the system satisfies the additivity property.

3. Scaling property:

If the input is scaled by a constant factor and the output is also scaled by the same factor, then the system is said to be scalable.

Let us apply a scaling factor of a to both the input and the output:

y1[n] = n*a*x[n]
y2[n] = a*y[n] = a*n*x[n]

Since y1[n] is not equal to y2[n], the system does not satisfy the scaling property. Therefore, the system is not linear.

Conclusion:

Based on the above analysis, we can conclude that the given system is non-linear.