Linear systems are defined by their adherence to what is called the superposition principles.1 There are just two superposition principles and they are called homogeneity and additivity. Firstly additivity, which states that we can add the effect or output of two systems together and the resulting combined system will be nothing more that the simple addition of each system’s output in isolation. So for example, if I have two horses that can each pull a hundred kilograms of weight on a cart in isolation, well if I then combine these two horses to tow a single larger cart they will be able to pull twice as much weight. Another way of stating the additivity principle is that for all linear systems, the net response caused by two or more stimuli is the sum of the response which would have been caused by each stimulus individually.2 The second superposition principle, homogeneity, states that the output to a linear system is always directly proportional to the input. If we put twice as much into the system we will, in turn, get out twice as much. For example, if I pay 50 dollars for a hotel room, for which I will get a certain quality of service, this principle states that if I pay twice as much I will then get an accommodation service that is twice a good. When we plot this on a graph we will see why linear systems are called linear because the result will always be a straight line.
These principles are of course deeply intuitive to us and will appear very simple, but behind them are a basic set of assumptions about how the world works. So let’s take a closer look at these assumptions that support the theory of linear systems. Essentially what these principles are saying is that it is the properties of the system in isolation that really matter and not the way these things are put together or the nature of the relationships between them. This is very abstract, so let’s illustrate it with some examples. Imagine you have some ailment and you have two drugs that you know are meant to cure this problem, so you take them both at the same time. The result of this, or we might say the output to this system, will depend on whether the two drugs have any effect on each other when taken in combination. If the drugs have no effect on each other then it will be the properties of each drug in isolation that will define the overall output to the system, and because of this lack of interaction between the components our linear model will be able to fully capture and describe this phenomenon.
But if the drugs do have some effect on each other, then it will be the relationship between them that will define the system. And our linear model that does not account for this will fail as it is based upon the principle of additivity that assumes a simple additive relationship that is not the case in this situation. So we can then see the basic reasoning behind additivity, that we can simply add things without any regard to how they will interact when we put them together. Now behind the principle of homogeneity is the assumption that scale does not matter. So say we have a business producing a million widgets a year and then scale this up to producing two million the next year. Well, maybe everything will simply scale in a linear fashion that will, of course, be captured by our linear model, but also it may not. If I can leverage economics of scale then costs will not grow in a linear fashion, and if by producing twice as much I saturate the widget market, then this will feed back to reduce my revenue resulting in a scaling that is not linear in nature.
Linear systems models fail to capture feedback. They do not take into account the effect that an action of a system will have on its environment and how that will, in turn, feed back to affect the system again, not just in space but also in time, that is to say how past actions will feed in to affect the current state of the system. The model of a linear system essentially exists in a static time vacuum. One might wonder why we use linear systems models at all if they fail to capture so much of the real phenomena that we see in the world around us. But there are a number of good reasons why we do. Firstly, linear models are deeply intuitive to us. The static properties of real tangible things that linear systems theory captures is much easier for us to see, touch and quantify as opposed to the intangible world of the relations between these things and over time. Secondly, as we have noted linear models do capture the behavior of some if not many systems, such as the simple interactions between particles of matter or simple dynamics of cause and effect that we might sometimes see in social and economic behavior.
Lastly and probably most significantly, linear models are inherently simple. They also remove any qualitative questions surrounding the nature of the relations between elements in the system. This makes them particularly amenable to the rigorous quantitative methods of mathematics and the reductionist approach to science, where we can approach complex problems by breaking them down into their constituent parts and then tackle these simpler problems in isolation.