What is the purpose of linear algebra in real life?

[Moonbear]

Okay, I've had this question lingering on my mind for a LONG time, so I'm finally going to ask. I took a semester of linear algebra WAY back when I was in college; I think it was one of the chem major requirements when I still thought I was going to be a chem major. I aced the class, but there was one key thing I never learned and was never explained to us...just what is the purpose of linear algebra? I mean, it was fun solving all those problems with matrices, but I never had a clue what the purpose was! What are the applications of linear algebra? What sort of things would those numbers in matrices represent in real life (you know, outside of a textbook)?

 

[Palindrom]

I'm not willing to believe it's not being used in Biology.

Anyway, Linear Algebra is being used in so many places, both in Physics and Mathematics, and I am told also in Computer Science and Electrical Engineering...
Any set of linear equations in Physics and Mathematics is solved with Linear Algebra, and it's only the simplest use of it! It also comes in hard in P\ODE.
In Quantum Physics, for example, all is formalised to use Linear Algebra by nature. For example, measurable things (like x, or the momentum P, the linear momentum L, the energy, usually the Hamiltonian H) are described by hermitean operators, and the only things you can measure in experiments are eigenvalues of these operators!
And there are many, many other uses to it. I've come to find out almost every Mathematical branch is used in Physics here or there.

 

[shmoe]

As an application perhaps a biologist can appreciate, you can use best approximations to linear systems to find the least squares line (or polynomial of any degree, or actually linear combination of any set of functions) to a given data set.

Another handy thing is you can model rotations with matrix multiplication. One of the first things I did after my first linear algebra course was make a lame spinning and bouncing box screen saver on my computer. This was a great improvement over my lame bouncing box that didn't spin. You can imagine my increase in popularity.

 

[matt grime]

Linear algebra is the start of many things biological - mathematical modelling of disease epidemics being the most significant. Of course this is just one ingredient. As you did chemistry, briefly, then linear algebra (well, it's natural extension) can be used to work out the normal modes of vibration of atoms in molecules.

Linear algebra is the study of the way a system such as the real numbers, or the complexes, or the rationals or any other field can act on multidimensional objects. The natural generalization is to study things more complicated than fields (groups, initially) and how they act on "spaces". These are the natural mathematical ways to think about many phenomena from elementary particles right on up to the geometry of the universe.

 

[gravenewworld]

Linear algebra is the start of many things biological - mathematical modelling of disease epidemics being the most significant. Of course this is just one ingredient. As you did chemistry, briefly, then linear algebra (well, it's natural extension) can be used to work out the normal modes of vibration of atoms in molecules.

Speak of the devil. Right now I am studying for my final exam in inorganic chemistry which includes group theory and matrix algebra in order to predict which stretches of a molecule will be active in IR and Raman spectroscopies.

 

[Moonbear]

Speak of the devil. Right now I am studying for my final exam in inorganic chemistry which includes group theory and matrix algebra in order to predict which stretches of a molecule will be active in IR and Raman spectroscopies.

Hmm...maybe the problem wasn't with the math course, but with the chemistry courses. We never used linear algebra in the inorganic chem course I took. I think I took linear algebra after inorganic chem (or maybe at the same time), so don't even think it was considered a pre-req for inorganic.

Okay, I appreciate all the responses so far, but apparently my recollection of linear algebra is even hazier than I thought . I seriously have no idea what anyone else was talking about. Truthfully, the course I took must have been pretty pitiful...how can someone get an A in a decent course when they don't even understand the purpose? Oh, except modeling rotations. That's something I can wrap my brain around a bit. So, how does multiplying matrices translate into a rotating model? I think there's something very fundamental missing in my understanding of the subject.

Palindrom, I don't know if there are fields of biology that use it, but it was never a required course for a bio major (bio majors only needed 2 semesters of calculus, not even multivariable calc). It certainly isn't something I use, though if it's involved in model rotation, it probably is used to create some of the software I use.

 

[shmoe]

Oh, except modeling rotations. That's something I can wrap my brain around a bit. So, how does multiplying matrices translate into a rotating model? I think there's something very fundamental missing in my understanding of the subject.

Do you remember linear transformations from one vector space to another? Basically a function T from $$\Bbb{R}^n$$ to $$\Bbb{R}^n$$ is a linear transformation if for all vectors $$u,\ v\in\Bbb{R}^n$$ and any scalar $$a\in\Bbb{R}$$ we have $$T(u+av)=T(u)+aT(v)$$. The idea is a linear transformation 'plays nicely' with vector addition and scalar multiplication- if you have two vectors, add them, then transform them you get the same result as if you transformed them each separately then added them. Similar deal for scalar multiplication, it doesn't matter if you multiply a vector by a scalar then apply your transformation or if you apply your transformation to the vector then multiply the result by a scalar.

To keep things simple, think of a rotation about the origin by $$\pi/2$$ radians counter-clockwise in the plane, $$\Bbb{R}^2$$. It turns out this is a linear transformation. Think of your favorite vector, rotate it by $$\pi/2$$ radians, then multiply it by 2 (just stretches it). Now think what happens if you were to multiply your favorite vector by 2, then rotate it- same result. Similar dealie with vector addition (think of the parallelogram law for vector addition if you like).

Now it turns out that all linear transformations from $$\Bbb{R}^n$$ to $$\Bbb{R}^n$$ are just matrix multiplication (and vice versa). That is, if T is a linear transformation, there is an nxn matrix A where T(u)=Au for all vectors u. For our rotation by $$\pi/2$$ counter clockwise, this matrix will be

$$A=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$$

If we wanted to rotate the vector $$\begin{bmatrix}1\\1\end{bmatrix}$$, we'd just look at the product $$A\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}-1\\1\end{bmatrix}$$, which is what we'd expect, only with some ugly formatting.]

You can test this out on your favorite vectors in $$\Bbb{R}^2$$ in the comfort of your own home to see that it does actually perform as advertised. Without too much difficulty, you can produce the matrix that gives rotations by any angle we like. We can also produce rotations in higher dimensions for added fun. Hope that sheds a little light on rotations.

 

[Gokul43201]

In two dimensions, to rotate any vector $$\vec{x}$$ through an angle $$\theta$$, you pre-multiply it by the rotation matrix
$$R = \begin{bmatrix}cos \theta&-sin \theta\\sin \theta&cos \theta\end{bmatrix}$$

In 3 dimensions you can create a 3X3 matrix by composing rotations about each of three axes.

 

[matt grime]

You asked what linear algebra was used for. I hope you didn't expect the uses to necessarily make sense to someone who;s done one course in it: would you expect someone with a single course in, for example, selective plant breeding to be able to explain the distributions of codons and how many chromosomes humans have? Ok, it's a poor example but it's early here and I can't quite find the right one.

Linear algebra is a basic tool, a simple and well understood bit of mathematics that anyone can understand.

Here's some direct linear algebra to understand.


Let L(n) be the population in year n of lions, Z(n) the population of zebras. It is not unreasonable (well, ok it may be) to say that the population in year n+1 will depend proportionally on the previous two years in a typical hunter-prey model, then if we presume the constants behave as:

L(n+1) = aL(n)+bZ(n)
Z(n+1)=cL(n)+dZ(n)

and so on, then we can try and find under what conditions the populations stabilize and such.

Actually the system is more complicated, and we ought to introduce random factors and so on, but it's a good start and I think you can see that linear algebra will be of direct use.

 

[ComputerGeek]

the study of species deals with Linear algebra. when one wants to measure the long term viability of a population, one needs to use eigenvalues, eigenvectors, and eigenspaces.

you are telling me that you never used these tools to study animal populations?

 

[JasonRox]

My question is:

Why does everyone question the usefulness of Mathematics?

I'm actually sick and tired of it.

 

[Gokul43201]

My question is:

Why does everyone question the usefulness of Mathematics?

I'm actually sick and tired of it.

Irrelevant.

No one in this thread has questioned the usefulness of mathematics.

 

[fourier jr]

i'm fed up with people questioning the usefulness of math too but ya, nobody asked that here. (doesn't everyone get asked if they're just going to teach?? i'll go postal if just 1 more person asks me that...)



here's a multiple-species example of the Lotka-Volterra model from my old modelling book:
N_1, N_2, N_3, N_4 are 4 species such that N_2 preys on N_1, N_3 preys on both N_1 & N_2. assume that N_4 in this instance has a symbiotic relationship with N_2 & is neutral wrt the other two. the resulting SYSTEM OF EQUATIONS (!) is:

$$N_1' = c_1N_1(1 - \frac{N_1}{K}) - d_1N_1N_2 - e_1N_1N_3$$

$$N_2' = c_2N_1N_2 - d_2N_2 + e_2N_2N_4 - f_2N_2N_3$$

$$N_3' = c_3N_3N_1 + d_3N_3N_2 - e_3N_3$$

$$N_4' = c_4N_2N_4 - d_4N_4$$

in the system, K is the environmental carrying capacity & all the c's d's, e's & f's are real-valued constants.

applied math = x 100

(it's been a while since i really looked at the chapter but it looks like it's all about systems of odes & a linearization procedure to make things nicer)

 

[Moonbear]

Irrelevant.

No one in this thread has questioned the usefulness of mathematics.

Correct. I was addressing my ignorance of the use, not whether there is a use. I'm confident there are uses and that it is indeed useful; I was just trying to close a gap in my own knowledge.

The rotation example was very helpful. That makes sense.

Matt Grime, thanks too! Yes, I didn't expect to have mastered any sort of real applications out of a single class taken *mumble*[/size] years ago, but with all my other classes, at least some practical examples of uses were given, even if they were simple examples.

But, you know what, don't sweat it folks. In the interim since my last post, I've gone back to my old textbook to refresh my memory. There's a bookmark still in it from the last chapter we covered in the class (can you believe it!)...I seemed to recall we hadn't gotten through the entire book and had covered chapters at a painfully slow pace...sure 'nough, we never got past Chapter 3! I'm pretty sure I got gyped out of a decent class (I should ask for my money back, huh? ). Chapter 4 covers eigenvalues. No wonder I'm so confused by what you all are talking about! (And no, this wasn't a class intended for bio majors! Good heavens! I was in a class with engineers and math majors...they must have really struggled to catch up in their more advanced classes!)

Like I said before, I don't know if other biologists use this stuff, but I haven't had a need for it. I don't study population biology, which are all the examples given. Then again, in biology, they tend to teach you equations as you need them, out of context of any formal math course (unless you're going into mathematical modeling), so folks may use this and not even know what field of math it formally fits into. All the math I use on a regular basis was covered in elementary school. I count a lot of cells!

Despite wandering into this topic out of nearly complete ignorance, this has been helpful. Thanks folks!

 

[theFuture]

Markov chains come to mind first.

 

[mathwonk]

as my 12 year old son said when i asked him if he needed algebra to compete in the math contest: "Well dad, you don't really need it, but if you know it, you can use it."

 

[Nereid]

So far the examples given have related primarily to theory.

There's also the experimental side - if data from the lab or field is quantitative (and I would expect that much of it is in modern biology), then it will likely be analysed with some general (stats) package or special application ... if you poke around inside these, I think you'll find 'linear & abstract algebra' being put to very good use.

 

[ComputerGeek]

So far the examples given have related primarily to theory.

There's also the experimental side - if data from the lab or field is quantitative (and I would expect that much of it is in modern biology), then it will likely be analysed with some general (stats) package or special application ... if you poke around inside these, I think you'll find 'linear & abstract algebra' being put to very good use.

probably a lot of probability Matrices.

 

[Moonbear]

So far the examples given have related primarily to theory.

There's also the experimental side - if data from the lab or field is quantitative (and I would expect that much of it is in modern biology), then it will likely be analysed with some general (stats) package or special application ... if you poke around inside these, I think you'll find 'linear & abstract algebra' being put to very good use.

Cool! I'm probably using it and don't even know it. I don't know a lot about how many statistics are computed. My statistics courses focused on ANOVA and linear regression/orthogonal contrasts. The emphasis was on designing the experiments, though we did have to learn to compute an ANOVA table by hand. I don't know how nonparametric statistics are computed...when I need those, I go ask a statistician for help; that's what they're there for!

 

[cronxeh]

Well IMHO, the most beautiful things come to life when you combine Linear Algebra with Differential Equations. Also nonlinear diff eqs with its applications in statistical and quantum mechanics, electromagnetism, population dynamics, chemical kinetics, combustion theory, heat flow, wave motion, vibrations, poisson equations, and many more 'real world' problems can be solved with linear algebra + diff eq methods.

There are enormous applications to computational biology in this. From bioinformatics to neuroscience to systems biology - if you want to advance forward you will eventually stumble upon the heavy math and computer usage

 

[JasonRox]

as my 12 year old son said when i asked him if he needed algebra to compete in the math contest: "Well dad, you don't really need it, but if you know it, you can use it."

You have one wise kid.