# What good is linear algebra?

The right question to be asking is: "what good is any math other than linear algebra?"

mathwonk
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2020 Award
in short, linear problems are the only kind we can solve. thus most other math subjects consist in giving the best possible linear approximation to a given problem.

e.g. we can solve linear diff eq's, hence we try to give linear approximations to non linear dif eq's to at least conclude what kind of solutions exist, and to approximate those solutions.

Would it be correct to say that Lie algebras are like linearised Lie groups?

mathwonk
Homework Helper
2020 Award
now you are catching on.

I heard dot products are very good for physicists especially when it comes to calculating stuff like Work.

There are many applications of linear algebra in physics such as:
1. Solving circuit problems with Kirchhoff's law.
2. Control theory where you need to determine whether the system is controllable or observable.
3. Stablity of systems where you need to find the poles of the system.
4. Finding the period of vibration of a building.
5. Heat transfer problems.
Many more areas. If you want to find more applications try the following:
http://www.palgrave.com/science/engineering/singh/ [Broken]

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There are many applications of linear algebra in physics such as:
1. Solving circuit problems with Kirchhoff's law.
2. Control theory where you need to determine whether the system is controllable or observable.
3. Stablity of systems where you need to find the poles of the system.
4. Finding the period of vibration of a building.
5. Heat transfer problems.
Many more areas. If you want to find more applications try the following:
http://www.palgrave.com/science/engineering/singh/ [Broken]

In Chemical Engineering, determining whether production levels can be met in a refinery.
e.g. no solutions if production levels cannot be met.

In Economics, determining the supply and demand of certain products.
Negative numbers mean imports must be required to sustain the product or consuming other products quickly to cool down the in most demand.

I can use it to determine if I can spend a certain amount of money overseas in Asian countries and see whether I can pay for all food, hotel, shopping and other expenses.

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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

Well, the real world is quantum mechanical and quantum mechanics is linear algebra on a Hilbert space. So, all real world problems can be reduced to linear algebra.

If you would like to learn and understand representation theory, linear algebra/matrix theory is essential.

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HallsofIvy
Homework Helper
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
Homework Helper
2020 Award
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
Homework Helper
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
Homework Helper
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.

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!!