Where can I find Linear Algebra in Nature?

I'm a Computer Science major and I've been studying Analytic Geometry and Linear Algebra this semester. Today my teacher gave a hell of an explanation talking about linear systems, quadratic functions, polynomial equations, derivatives, etc., etc.. In his examples, he talked about Ballistics, about space and all matter, about the physical "tissue" that covers every object, cars and its aerodynamics, and so on and I was basically tripping with so many crazy ideas.

I really, really loved this subject and have even been considering studying it more out of the Academia for curiosity purposes.

The thing is, I couldn't find more of these examples where linear algebra can be used in nature. I know about some possible uses in Computer Science such as Computer Graphics, A.I., cryptography and etc..

Can anyone shed a light?

Thanks!

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You'll probably get manny different exciting answers, but there's a real challenge there, because to most people, linear algebra has become so natural, that they don't even think about it when they are using it. – Olivier Bégassat Aug 16 '11 at 2:49
Possibly not many places. Very fine-grained models of nature seem to be largely non-linear. However, linear approximations are enormously effective. – André Nicolas Aug 16 '11 at 3:12
In a similar vein as André's comment: a lot of stuff in nature happens in a continuous manner (for which mathematical modeling associates so-called "operators"); what we do to study these models is to discretize these operators. The operation of turning the continuous into the discrete is what gives rise to matrices... – J. M. Aug 16 '11 at 3:23
When I take a shower, I always think of adjusting the water temperature and volume as solving a linear system of two equations in two unknowns (the unknowns being the angles through which I turn the hot and cold water knobs). – Gerry Myerson Aug 16 '11 at 3:35
To illustrate André's comment, you may for example think about computer graphics, where arbitrary shapes are approximated by polygons and that's where linear algebra comes in. – Joel Cohen Aug 16 '11 at 3:52

Okay. There are loads of ways linear algebra can be used to look at aspects of nature. It's a tool, just like calculus, and thus can be used in many, many, many different ways. I will speak about 6: 3 relatively simple applications and 3 not-as-simple (or straightforward, at least) applications.

Systems of linear equations are sort of the bread and butter of linear algebra. So studies like chemistry, which involve (as we now know) discrete reactions of a certain amount of moles of some substances into new substances. It's actually nontrivial, as no chemical reaction occurs completely, and no measurement is taken perfectly. So finding out, for example, that combusting 1 part Methane in 1 part Oxygen returns two parts water, one part carbon dioxide, and some heat is true - but all observations of such things will be slightly off. So least squares can be used, too. Less trivial still is the idea that multiple substances are dissolved in different water containers, and then combined. If this is repeated with different concentrations, then one can use linear algebra to solve for both the volumes of all the solutions used and the chemical formula of the resulting compound in terms of the solution's compositions (this is a bit detailed, so I don't include an example).

This also has big implications in economics. In the so-called Leontief Closed Model of economics, there is an assumption that during the production period being examined, no goods leave or enter the system. Thus the numbers of goods produced and used form a closed system, and everything adds up pretty and linearly. Then the idea is that one can measure and predict the balance of the economy by knowing how many units of each good are produced by industries and how many are needed to produce other goods. The basic idea is that one compiles a matrix containing the necessary inputs and resulting outputs of goods for different industries (imagine each row is a different industry, or company, or producer, etc. and each column represents a different good, i.e. the basis is over the different products - and they're not orthogonal, as some products add together to give other products). Then one needs to be able to multiply this matrix by a goods-column vector and get that vector back for the system to be balanced. If it's not, then there are big problems (unbalanced systems indicate an unsustainable economy).

But there is more, as the Leontief Open Model, where goods are allowed to enter and exit the economy, can also be analyzed. A simple way of looking at it is to modify the above system. If $A$ is my producer/good matrix, and $p$ is my product vector, then the above equation was $Ap = p$. But we might introduce a demand vector $d$, indicating entrance or exit of goods. Then we look at $Ap + d = p$. This allows one to analyze the levels of production of the economy and many other aspects. Again, there are more complicated formulations with linear algebra, but I want to give breadth.

I will give one more relatively basic implication. Genetic trends can be efficiently modeled with matrices. In particular, if we know dominant traits and the number of genes affecting a genotype, then we can form a vector out of the estimated probabilities of each genotype occurring (probably through some sort of statistical sampling). If we form an outer product from this vector (make a matrix, of sorts), we have a probability matrix for the genotypes offspring of two parents. Finding eigenvalues allows one to analyze long term behavior. In addition, similar estimates and probability matrices allow us to infer past behavior.

Those were three lower-level applications (in my mind - nothing rigorous about these definitions). Now for three upper-level applications.

Oscillators - anything resembling harmonic motion - are often governed by linear systems. Why? Because of Hooke's Law - that says that the force of a spring is proportional (i.e. linear) to its displacement. One oscillator, though, is an intro mechanics question. But coupled oscillators are, to be honest, hard (in my opinion - some physicist is rolling his eyes at my answer now). For example, the configuration where we have a wall, a spring, a body, a spring, a body, a spring, a wall - each spring are identical and the track is frictionless - this system is governed by a linear set of equations. They are perhaps a little bit clever, but one can use eigenvalues to understand this system. In a more general manner, systems of differential equations are often understood through hefty amounts of calculus and linear algebra - so less trivial systems can be understood too (the goal is still to find the eigenvalues, too).

Sociologists are interested in analyzing relations inside of groups - group behavior. Sometimes, to do this, they make graphs (in the graph theory sense, not in the function-plotting sense) where each vertex represents an individual or a particular group. The edges between individuals can be given a weight and a direction (or perhaps a bidirection) representing influence or dominance (I'm not entirely sure how this is done, but I know it is - I suppose you'd have to ask a sociologist). Then one can make adjacency matrices (a matrix reflecting what individuals affect what individuals) and do funny things with them - squaring them will show what individuals have a long influence on the group. Cubing will show even longer influence, etc. The underlying concept here is that networks can often be modeled my graphs, and graph theory can use a lot of linear algebra. So communication networks, electric circuits, even terrorist networks (picture below) can often be examined with linear algebra. I want to note that in the network pictured below, when one computes powers of the adjacency matrix, all but 10 individuals influences die down almost immediately. But these 10 are all relatively equal in influence, so this is called a non-centralized network.

If you've read this far, then I'm proud. This is far longer than I had anticipated. But so it goes. The last idea I want to mention is that linear algebra is not restricted to only basic results. I hope I have given that impression, but here is one more. There is a paper that explains how to derive the Lorentz transformations with linear algebra. Like most physics, it's a close approximation. I think the paper is interesting to those with only a basic familiarity with linear algebra, though some things might feel a little mystical.

Anyhow, I hope you found this interesting.

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Wow. Just wow. This is far more than I expected to get as an answer. Everything you said was very amazing, the genetic trends, the Leontief models and the group behavior analysis... I find it beautiful how a mathematical model apparently simple can explain so much. And this reminds me of an article from the last Scientific American about wether Maths was invented or discovered, which is a whole other discussion, but anyway, thank you, very, very much. I'm very amazed by your examples, by the trouble you went through to write this, and by the paper on Special Relativity and Linear Algebra. :) – Bernardo Oliveira Aug 17 '11 at 2:59
Great answer. One example of a sociological application is Google's PageRank algorithm. en.wikipedia.org/wiki/PageRank – Dan Brumleve Aug 17 '11 at 5:51

Another idea is to look at nature statistically. If you want to describe the joint effect of various forces/influences on some measurable effect, where every part of your observation and your theoretical model is quantified with a metric of real numbers - then the decomposition into the parts of each influence is very natrually done by means of linear algebra. For instance multiple regression, factor analysis and canonical correlation are cases, where such a model is basically thought as explanative for combined "influences"/forces.
Well, it's arguable, whether "linear algebra occurs in nature" this way- linear algebra is just a way to analyze the observed nature (as far as we have mathematical models of its interrelations)

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Depends on what you mean by "nature". In the most fundamental level, quantum mechanics is all about linear algebra.

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And here I was thinking it was about Differential Equations! (I didn't downvote, btw... I always kiss and tell) – The Chaz 2.0 Aug 16 '11 at 3:05
That linear algebra apparently appears in nature is due to the fact that we barely understand nature and linear models are a good first approximation. In computers, linear algebra drives graphic engines. So much so, that it is told that matrix multiplication is in hardware rather than software. – Scott Carter Aug 16 '11 at 3:12
Well, I meant as everything that's natural in it's essence. Could be planets, plants, bodies and forms, the way some things are the way they are or simply the way an apple falls from a tree. And thanks for the tip, I'll definitely take a look on Quantum Mechanics. :) – Bernardo Oliveira Aug 16 '11 at 3:12

Anywhere differential equations or chaotic phenomena appear, linear algebra appears in nature, Be. The solutions of any linear differential equation, ordinary or partial,of any type forms a vector space over the real or complex numbers. That alone would mean linear algebra appears in most applications of mathematics to the physical sciences,but it's actually more profound then that. It turns out in principle, any asymptotically stable nonlinear differential equation system can be approximated to within a nieghborhood of it's solution by the solution of a linear one. This means linear algebra is present at some point in the solution process of virtually any system modeled by a differential equation or a system of differential equations. This is of particular interest to a computer scientist since many innovative algorithms have been developed in order to generate the accuracy of the approximation accomplished by today's supercomputers.

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I think someone's trying to leverage me off the site-every time I post a comment,my rep goes down. I never said it was a BRILLIANT observation,people. But the person asking the question IS an amateur,after all.I'll edit the post to be more informative when I have time. – Mathemagician1234 Aug 16 '11 at 5:31
Perhaps the downvote is for inaccuracy of the answer. You say that the solution set of any differential equation forms a vector space, but this seems not to be true already for e.g. $\frac{dy}{dx} = 5$. – Pete L. Clark Aug 17 '11 at 18:33
@Pete-I said the solution space of any LINEAR differential equation forms a vector space,Pete.And isn't the solution of the separable equation you gave 5x +C, where C is a real number? That's a set of linear equations,last I checked-what am I missing here? – Mathemagician1234 Aug 18 '11 at 18:06
The differential equation I gave you is linear: it is not homogeneous. And the family $5x+C$ is not a vector space in any natural way: the sum of two such functions is not another such function. (You can fix what you wrote either by restricting to homogeneous linear equations or by changing "linear space" to something like "torsor under a linear space".) – Pete L. Clark Aug 18 '11 at 19:10
@Pete You're correct about the homogeneousness-my bad.It's an easy mistake to make when you haven't studied elementary ODEs for awhile because you get used to thinking of linear nonhomogeneous differential equation solutions as the "linear homogenous part plus something else"-and it's easy to forget that's not necessarily a vector space! – Mathemagician1234 Aug 18 '11 at 20:58

A short description of Quantum Information Theory is the study of unitary and related operators on tensor products of Hilbert spaces (complete inner product spaces). This is not, as Scott Carter incorrectly indicates in the comments, because quantum mechanics is just a linear approximation. (It is true that the standard model of quantum field theory isn't rigorously understood in a nonperturbative sense, but that is a problem of mathematical analysis not related to the basic framework for all such theories as postulated by the Wightman axioms.) A good introduction to this subject is John Preskill's course notes from the quantum computation class at Caltech. (Just do a google search for preskill notes physics 219 if you're interested.)

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I basically have no knowledge of quantum mechanics, but as I understand it, the study of quantum mechanics heavily uses linear algebra. PlanetMath indicates general relativity does also.

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Thanks! That link to PlanetMath gave me a few more ideas! It's amazing, really. I'll do some research about its role in quantum mechanics too. Thank you. – Bernardo Oliveira Aug 16 '11 at 3:10