What are the real-world applications of linear algebra?

[HallsofIvy]

You hit the nail on the head with

"From my (limited) experience, it seems that the truley interesting and meaningful things in this world are rarely linear"

It's true. There are no interesting linear problems to solve, because we already know how to solve linear problems. And also there are no relevant linear problems to solve, because they've already been solved.

My question is" "Does that mean it's not worth learning how to solve linear problems?"

You should try to solve interesting, real-world problems without linear algebra, and then come up with a response to my question

The best way to approach non-linear problems is to approximate them by linear problems, then perhaps use the solution to that linear problem to get a better approximation.

For example, Newton's method for solving non-linear equations involves approximating the function at a given starting point by a linear function, then using that to get another starting point for another linear approximation, etc.

In quantum mechanics, the WKB approximation uses successive linear approximations to the non-linear differential equation.

 

[mathwonk]

halls tells it exactly as it is. it is because linear problems are the only ones we can solve, and yet most problems are non linear, that calculus was invented. differential calculus is the science of approximating non linear problems by linear ones.my recently posted notes for math 4050, on my website also illustrate the use of linear algebra in linear differential equations.

it turns out that every finite dimensional linear operator over C, is a direct sum of copies of the operator D = differentiation, acting on suitable spaces of smooth functions. this fact is called the jordan form.

i.e. if T is any linear operator on a finite dimensional space, with minimal polynomial f, then T has a matrix representation whose blocks are all representatives of D acting on the space of solutions of a differential equation with characteristic equation dividing f.

thus not only is linear algebra prerequisite for understanding linear diff eqs, but linear diff eqs are prerequisite for motivating jordan forms. the two subjects go hand in hand.

 

[Defennder]

thus not only is linear algebra prerequisite for understanding linear diff eqs, but linear diff eqs are prerequisite for motivating jordan forms. the two subjects go hand in hand.

It's good to know I'm taking my classes in the correct order.

 

[morphism]

There are no interesting linear problems to solve, because we already know how to solve linear problems. And also there are no relevant linear problems to solve, because they've already been solved.

Of course this is completely false.

 

[mhazelm]

You probably already have enough replies to this, but ;)

I was also bored in my first linear/ODE course. But the following year I took an upper level class called "Theory of Linear Algebra" and that turned everything around. It's kind of like anything else in math -first you learn some of the basics of the mechanics of the problems - how to multiply matrices, etc. Kind of how you learn to do calculus problems before you learn how to prove calculus theorems.

If you take another linear course later, on the theory, you will find it very interesting (if you are interested in math). The abstractness can be a bit weird at first, but how cool is it to create these abstract spaces that just live in our mind (so to speak ) and yet have so many beautiful, elegant applications in the physical world?

And if you're asking yourself why you'd ever use it, just ask about physics - it's EVERYWHERE. Classical mechanics takes place on a vector space. E&M involves vector fields and tensors (these are multi-linear objects). Quantum has linear algebra all over the place, with Hermitian operators and eigenvalues. Even just the basics of waves involves linear - you'll use inner products all over the place without even realizing it. Linear is also a part of the foundation of differential geometry - you need the notion of inner product to begin discussing distances and what distance means; you can't begin to discuss things like curvature without some knowledge of the linear algebra's inner product.

I believe there is also use for it in the financial world, but I don't have specific examples off the top of my head.

My point is, yes, learning the mechanics of how to compute things - it can be boring. But once you move to the theory, and you begin to learn why you were computing it, and what this means in the bigger picture (i.e. how it applies to things you never expected it to), linear algebra is absolutely fascinating.

Anyway, I hope you take a more exciting advanced course someday!