Proving Linearity: x(t) -> y(t)

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In summary, the conversation discusses how to prove mathematically if a system is linear or not. The criteria for a system to be linear are that it must obey proportionality law and superposition. Two systems are given as examples, and it is shown that for a system to be linear, the output should follow the equation y(t)=a1*y1(t)+a2*y2(t). However, in one of the systems, the output does not follow this equation, indicating that the system is not linear. The conversation also includes a correction for a previous mistake.
  • #1
tronxo
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how can i prove mathematically if a system is linear or not? i mean, i know the system must obey proportionally law and superpositon, but i don't know how apply into it.
well, if anyone could help me, the systems i need to prove are:
x(t) -> y(t)= Cx(t) + k
x(t) -> y(t)= ∫ (from minus infinite to "t") x(e)d(e); where "e" is a dummy variable
 
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  • #2
look if i/p x(t) is scaled by a1, then
o/p is y1(t)=a1*Cx(t)+k

if scaled by a2 it becomes
y2(t)=a2*Cx(t)+k

now to obey linearity if i/p is a1*x(t)+a2*x(t), then o/p should be
y(t)=y1(t)+y2(t)

but in this case if i/p is a1*x(t)+a2*x(t)
y(t)=a1*Cx(t)+a2*Cx(t)+k

which is not equal to
y1(t)+y2(t)= a1*Cx(t)+a2*Cx(t)+2k

So this sys is not linear
 
  • #3
Sorry made mistake

ratn_kumbh said:
look if i/p x(t) is scaled by a1, then
o/p is y1(t)=a1*Cx(t)+k

if scaled by a2 it becomes
y2(t)=a2*Cx(t)+k

now to obey linearity if i/p is a1*x(t)+a2*x(t), then o/p should be
y(t)=y1(t)+y2(t)

but in this case if i/p is a1*x(t)+a2*x(t)
y(t)=a1*Cx(t)+a2*Cx(t)+k

which is not equal to
y1(t)+y2(t)= a1*Cx(t)+a2*Cx(t)+2k

So this sys is not linear
Sorry made a mistake there it should have been

if i/p is x1(t) , then
o/p is y1(t)=Cx1(t)+k

if i/p is x2(t) it becomes
y2(t)=Cx2(t)+k

now to obey linearity if i/p is a1*x(t)+a2*x(t), then o/p should be
y(t)=a1*y1(t)+a2*y2(t)

but in this case if i/p is a1*x(t)+a2*x(t)
y(t)=a1*Cx(t)+a2*Cx(t)+k

which is not equal to
a1*y1(t)+a2*y2(t)= a1*Cx(t)+a2*Cx(t)+(a1+a2)k

So this sys is not linear:redface:
 
  • #4
Thank u, I finally got the right result...
 

1. What is linearity in the context of x(t) -> y(t)?

In this context, linearity refers to the property of a system where the output y(t) is directly proportional to the input x(t). This means that if the input is scaled or shifted, the output will also be scaled or shifted accordingly.

2. How do you prove linearity in a system?

To prove linearity, we must show that the system satisfies both the superposition and scaling properties. This involves evaluating the system's response to different inputs, including combinations of inputs, and comparing the output to the predicted result based on the properties.

3. What are the superposition and scaling properties?

The superposition property states that when multiple inputs are applied to the system, the resulting output is equal to the sum of the individual outputs when each input is applied separately. The scaling property states that if the input is multiplied by a constant, the output will also be multiplied by the same constant.

4. Why is proving linearity important in scientific research?

Proving linearity is important because many natural phenomena can be described using linear systems. By understanding linearity, scientists can accurately model and predict the behavior of these systems, which is crucial in fields such as physics, engineering, and economics.

5. Are there any real-world examples of a linear system?

Yes, many real-world systems exhibit linearity. For example, the relationship between force and displacement in a spring obeys Hooke's Law, which is a linear relationship. Another example is the relationship between voltage and current in an electrical circuit, which follows Ohm's Law and is also linear.

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