How do I determine the linear/non-linearity of this problem?

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Homework Statement

Determine whether the system is (a) linear, (b) time-invariant:

Relevant Equations

y[n] = T{x[n]} = x(t)
y(t) = cos(3t)x(t)

A system is linear if it satisfies the properties of superposition and homogeneity.

Superposition: adding the inputs of two systems results in the addition of the two outputs.
Ex) x₁(t) + x₂(t) = y₁(t) + y₂(t)

Homogeneity: multiplying the input by some scalar value is equal to the output multiplied by that same value.
Ex) ax₁(t) = ay₂(t)

After doing some internet searching, I also found this equation for superposition: f(a+b) = f(a) + f(b)

However, I don't know how to apply these rules to determine whether my equations are linear or not.
My attempt for determining SP:
y₁[n] + y₂[n] = T{x₁[n] + x₂[n]} = n(x₁[n] + x₂[n])

If my attempt at SP is correct, what is the next step? If it is wrong, what is the right way to write it, and then what is the next step?

 

[Ssnow]

What is the operator $T\left\{\cdot \right\}$ ? $n$ is natural ? $y[n]$ is discrete and $y(t)$ is continuous ? Can you define better this system before to start to prove the two conditions ?
Ssnow

 

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What is the operator $T\left\{\cdot \right\}$ ? $n$ is natural ? $y[n]$ is discrete and $y(t)$ is continuous ? Can you define better this system before to start to prove the two conditions ?
Ssnow

Thank you for your reply Ssnow,

The second equation is a separate problem; apologies for being unclear. We can ignore that.

So for now we are just dealing with the discrete problem. Unfortunately, I am unsure what the T{} operator is for. I would guess that it is an arbitrary value representing a function of a function; i.e. x[n] is a function of T. Again guessing, I think n is some arbitrary value; it could be a, or b, or whatever else.

 

[Mark44]

The second equation is a separate problem; apologies for being unclear. We can ignore that.

So this is the one you are concerned with?
y[n] = T{x[n]} = x(t)

I understand the first part, y[n] = T{x[n]}, but not why this would be equal to x(t).

Here T is some transformation that is applied to x[n] to produce y[n].

So for now we are just dealing with the discrete problem. Unfortunately, I am unsure what the T{} operator is for. I would guess that it is an arbitrary value representing a function of a function; i.e. x[n] is a function of T.

No, you have this backwards. y[n] is a function of x[n] via the transformation T. T is a transformation that operates on a sequence or function x[n]. It is not a value.

Again guessing, I think n is some arbitrary value; it could be a, or b, or whatever else.

Finally, if you are not given a formula for T, there is no way you can tell whether it is linear or time-invariant. Are you sure that the second equation, y(t) = cos(3t)x(t), isn't part of this problem?

For a transformation T, Tis linear if T(a + b) = T(a) + T(b) and if T(ka) = kT(a), where a and b are in the domain of T, and k is a scalar constant.

 

[Lapse]

So this is the one you are concerned with?
y[n] = T{x[n]} = x(t)

I understand the first part, y[n] = T{x[n]}, but not why this would be equal to x(t).

Here T is some transformation that is applied to x[n] to produce y[n].
No, you have this backwards. y[n] is a function of x[n] via the transformation T. T is a transformation that operates on a sequence or function x[n]. It is not a value.

Finally, if you are not given a formula for T, there is no way you can tell whether it is linear or time-invariant. Are you sure that the second equation, y(t) = cos(3t)x(t), isn't part of this problem?

For a transformation T, Tis linear if T(a + b) = T(a) + T(b) and if T(ka) = kT(a), where a and b are in the domain of T, and k is a scalar constant.

Thank you for your reply Mark,

I am positive that the second equation is for a different problem because it is delineated as (2); instead of being a part of problem 1. However, maybe we should switch over to the 2nd problem if we can't discover the answer to the first one.

So, do I start by creating y₁ & y₂ and add them together?

y₁(t) + y₂(t) = cos(3t)[x₁(t) + x₂(t)]

What do I do next?

 

[Mark44]

Thank you for your reply Mark,

I am positive that the second equation is for a different problem because it is delineated as (2); instead of being a part of problem 1. However, maybe we should switch over to the 2nd problem if we can't discover the answer to the first one.

So, do I start by creating y₁ & y₂ and add them together?

y₁(t) + y₂(t) = cos(3t)[x₁(t) + x₂(t)]

What do I do next?

You're still going at this backwards.
From post #1: y(t) = cos(3t)x(t)
For linearity, you need to show (1) that $y(t_1 + t_2) = y(t_1) + y(t_2)$ and (2) that $y(kt_1) = ky(t_1)$.
So for (1), $y(t_1 + t_2) = \cos(3(t_1 + t_2))x(t_1 + t_2)$. Does that work out to $y(t_1) + y(t_2)$?

It would be helpful if you showed us the exact wording of the problem. If it's a problem in a textbook, upload a photo of the problem. As presented in this thread, there seems to be a lot of missing information.