What is a linear system?

A linear system is a physical system responding to an external stimulation in a manner which is proportional to the amplitude of said stimulation.

Stated otherwise, it is the study of a class of systems characterized by the fact that their behavior can be modeled as a linear function:

$$f(x) = k·x.$$

Graphically, this means that if one plots how such a system $$f$$ responds to a variation of a variable $$x$$ the corresponding plot $$(x, f)$$ is a straight line.

One example is a spring. In the example of the spring, the variable $$x$$ is the amount of deformation of the system and f represents the corresponding increase in the force exerted by the spring as it is compressed by an amount $$x$$.

To put it yet another way, a system is said to be linear if the variation of its output is proportional to a corresponding variations of its input:

$$f(N·x) = N·f(x) \tag{homogeneity}$$ $$f(x+y) = f(x)+f(y)\tag{additivity}$$

properties known respectively as homogeneity and additivity. When both these properties are are displayed by a system it is considered a linear system or more formally it is regarded as satisfying the superposition principle :

"The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually."

https://en.wikipedia.org/wiki/Superposition_principle

For instance, if you go on a journey and travel 10 km every day, then the amount of km covered per unit time is linear (with respect to the amount of time which has elapsed).

Why is that important :

1. A whole class of physical systems can be regarded as linear in a first instance : pendulums, springs, etc. but also the propagation of waves in a medium.

2. This in turn is handy because linear systems translate to linear equations. Linear equations are simple to solve analytically. This means that if a system is linear, at least in a first order approximation, one can solve analytically the equations which govern its evolution and therefore one can tell a lot about a system if one knows it behaves linearly with respect to some variables.

3. Exemples of linear systems are:

• the response of a spring to stress.
• the oscillations of a pendulum.
• vibrations in an elastic medium (propagation of waves)
1. Historically, it is the realization of the fact that the oscillations of a pendulum depend solely on the length of said pendulum and not on its weight that allowed clockmakers to build reliable clocks with a rather simple technology. The same applied to oscillations of springs: this lead to the conception of the first watches or chronometers, which in turn allowed for a whole new era of sea travel, relying on the use a sextant and a chronometer, to become possible.

2. Another reason why linear systems play an important role in physics is Taylor’s theorem which states that in first approximation the response of most systems to a sufficiently small change in its parameters is linear is a first approximation (whether it is the vibrations of a guitar string or the response of the stock market to a small perturbation).

3. On the the other hand, it is also interesting to study how non-linear systems behave. They are more complex to model and their equations are more complex to handle; also they do not always have straightforward analytical solutions, in which case they can only be studied by computer simulations and by experimentation, however non-linear phenomena are at the root of all complex systems from life itself to climatic feed-back loops to modeling chaotic behavior of the heartbeats to modeling chaotic behavior of the stock exchange to modeling economic systems or the weather.

4. Last but not least, non-linear phenomena can be very deceptive, because they are inherently more complex and go against our instinct, a famous example of that is the formula of compound interests.

5. Beyond oscillators and waves, linear algebra also plays an important role in in computer science (ML and AI).

• You also want that $f(x+y)=f(x)+f(y)$ Jun 12 '21 at 3:38
• What do you denote by a period (.) in an equation? Jun 12 '21 at 8:26
• @BrianBorchers : done. Thanks. Jun 12 '21 at 11:32
• @Wrzlprmft : in that context, I meant a product between real numbers (not a dot product, scalar product). If it is not helpful, tell me and I'll just remove them. Jun 12 '21 at 11:35
• Excellent answer. I think it would be relevant to mention the superposition principle. Jun 12 '21 at 14:18

Imagine you have a force $$F$$ acting on your system, and it responds in some way. You can write equation of motion and find the response to $$F$$; in case of a pendulum it would be the displacement $$d$$ as a function of $$F$$ (and time, of course). Now, a linear system is such that if it has two forces $$F=F_1+F_2$$ acting on it, its total response to it will be the sum of the individual responses to each of the forces: $$d(F)=d(F_1)+d(F_2)$$.

• Nitpicking, but I think that this isn't the definition of a linear system, but rather one of its properties: namely, the superposition principle. Jun 11 '21 at 17:04
• No--it is the definition. Jun 11 '21 at 17:28
• @FabiosaysReinstateMonica hmmm.... do you mean the definition is with linear operators? 1. It might be different in math and physics, and 2. then what's the linear operator if not the one defined as where superposition holds? Jun 11 '21 at 17:45
• Check Serge Hulne's answer: he never mentions the superposition principle, just that the output is proportional to the input. Simplifying: the definition of linear is that its graph is a (straight) line, so $y=x$ is linear, but $y=x^2$ and $y=sin(x)$ are not. Then you define the superposition principle, and you find that it is valid for the former, but not for the latter, thus making it a property, but not part of the definition. Jun 12 '21 at 0:06
• Or, in other words: what I was saying is that, of the 2 requirements listed by Steeven, the second one is enough to define a linear system, and from that you can derive the first one as a property. But I admit I'm not sure how. It could simply be I'm wrong, which is after all what everybody is saying :-) Jun 12 '21 at 0:08

The other answers give good physical descriptions. Let me here add the mathematical definition as well for reference.

Mathematically, there are two requirements for a linear system - if both are fulfilled, then the system is linear, and vice versa if the system is linear then both apply:

$$f(a+b) =f(a) +f(b)$$

$$k\cdot f(a) =f(a\cdot k)\quad, k\in \mathbb{R},$$

where $$a$$ and $$b$$ are objects (vectors, matrices, scalars, functions etc.) of the relevant space. The first is often called stability in addition, the second stability in multiplication. Often a quick visual scan can determine when a system is non-linear - just look for squared terms or functions such sine, cosine, logarithms and the like.

• In the realm of linear systems, the term stability is typically used in reference to the asymptotic behaviour of the systems in response to certain stimuli. I would thus not use the term stability in the above. Jun 11 '21 at 16:46
• @MassimoOrtolano I agree, stability is a term used with reference to convergence and the like. In my native tongue (Danish) it is being used in this other connection as well very often. Jun 11 '21 at 18:49

Very briefly, a system of equations is linear if in an obvious way any scalar multiple of a solution or sum of two solutions is again a solution. Of course this assumes that solutions live in a vector space, so that scalar multiplication and addition of solutions makes sense. The condition is quite restrictive in that it only allows variable quantities to occur "to the power $$1$$", as that is the only power$$~k$$ for which $$(x+y)^k=x^k+y^k$$ holds; on the other hand there are still a lot of operations that can be used freely (like multiplication by any fixed matrix of derivatives of any order) and the stated property of solutions allows greatly facilitated reasoning about them.

I am not qualified to say how a "physical system" is defined in general, but I suppose that in any case it will give rise to a system of (differential) equations that govern the system, in which case the physical system is called linear if those equations are. Linear systems are so important that even for systems that are not linear, one often first studies an approximate system that is linear (often by assuming quantities are small and then ignoring powers higher than$$~1$$) to get some qualitative insight.