Many different things in math/physics that don't seem to be related to linear algebra can be classified as elements in a vector space, such as polynomials of degree $n$ for example. This can be done as long as all the elements follow all of the vector space axioms.
Another example from physics would be waves $y(x,t)$ that satisfy the linear wave equation $$\frac{\partial^2y(x,t)}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2y(x,t)}{\partial t^2}$$
Which (since $v=\frac{\omega}{k}$) has solutions in the form of $$A\cos(kx\pm\omega t+\phi)$$
Since any addition between the solutions is algebraic, the commutativity associativity laws hold and there exists an identity which can be added to any solution so as not to change it ($0$), and an additive inverse to every solution such that the sum of the element and its additive inverse is equal to 0.
The distributivity laws also hold for all real numbers, as well as the associativity law of multiplication. Any solution multiplied by 1 is also itself, satisfying the unitary law. The vector space is also closed under addition and scalar multiplication, since the addition of 2 waves yields another wave.
My question is: Is there any use for classifying waves (or any other sets of things in math/physics that at first glance may not be related to linear algebra) as elements of a vector space?
Best Answer
It gives you access to the big toolbox of linear algebra.
Most of the results from linear algebra are solely based on the vector-space axioms. You have a lot of theorems beginning with: “If $V$ is a vector space”. Thus, once you can show that your structure obeys the vector-space axioms, you can apply these results without further ado. (Of course, there are some results from linear algebra that have further constraints such as your vector space being finite-dimensional, but it’s not that these are secret.) Of course, you can also prove all the stuff you need from scratch and without ever referring to vector spaces, but that’s rather tedious.
It allows you to use an existing language / thought structure
Once you work sufficiently long with vector spaces, you develop an intuition for such things as dimensions, bases, orthogonality, etc. Showing that a structure is a vector space allows you to apply that intuition to that structure and quickly think about things. Relatedly, you can communicate about that structure with others much more easily. For instance, just consider the examples in the first sentence of this section. (If you wonder why language and thought structures are related: Having a vocabulary for something is crucial to effective thinking about it.)
This is the general way mathematics operates: You recognise similarities in structures, create an abstract framework that comprises these similarities, and benefit from synergy effects. For a blatant example, consider numbers: At some point, people realised that when you tie some strings together, their lengths behave in the same way that the time required for two subsequent activities behaves. Numbers allow you to think and speak about these things in a coherent way and you do not have to reinvent the wheel again every time you find something new to quantify.