Okay, that lets you visualize it (in the finite case) but it's a terrible way to sell it. Spreadsheets are booooring. Did you know that functions are vectors? Okay, better. Did you know that quantum mechanics is all about linear algebra? Okay, sold!
1. Almost any time you work in more than one dimension you will want linear algebra in your toolbox. There are a zillion methods for solving (non-linear) equations out there, and in more than one dimension, they use linear algebra. Newton's method? Incredibly useful in practice due (quadratic convergence rocks!), and with some linear algebra sauce BOOM you have Newton's method in as many dimensions as you can sneeze at.
2. Oh, by the way... did you know that the Fourier transform is linear?
3. Back to quantum mechanics... there's a thing you can do with a linear operator (a matrix is a kind of linear operator) where you get the "spectrum" of the linear operator. It's useful for making sense of big matrices. But in QM, the wavefunctions for electrons are described as eigenfunctions of a linear operator, and taking the "spectrum" of the linear operator gives you the actual spectrum of light that the chemical under study emits. Hence the name, "spectral theorem". It may be linear algebra on paper, but it's laser beams and semiconductors in the real world.
4. Oh hey, want to learn about infinite-dimensional vector spaces? Maybe some other time..
5. It's hella useful for modeling. Any model is wrong, but Markov processes are useful. Say you run an agency that rents out moving vans, and you have facilities in 30 cities. Vans rented in city A have a 10% chance of being dropped off in city B, 7% in city C, 9.2% in city D, etc. At this rate, how long till you run out of vans in city F? It's a differential equations problem with like 30 different equations! Or you could rewrite it as a single equation with matrices. You'll end up with weird things like 'e^(A*t)' where A is a matrix, and you thought "no way I can exponentiate to the power of a matrix" but spectral decomposition is like "yes way!" and you can solve the equation by diagonalization. Radical! (Basically, linear algebra rescues differential equations from the pits of intractability. I'm using rental vans as an example, but it could be a chemical reaction or a nuclear reaction or a million subway riders or whatever you want.)
So the question is:
Do you find economics, quantum physics, chemistry, engineering, classical mechanics, machine learning, statistics, etc. useful?
Then get some linear algebra in you!
- you've mixed in with nonlinear phenomena (most physical processes) with linear, and there's less and less reason to pretend linearity with increasing computer speeds (although in fairness you alluded to models being wrong)
- poo pooed spreadsheets without explaining why (you alluded to QM etc. but they don't care about useless proofs in linear algebra unless they're pencil pushing time wasters)
- implied infinite-dimensions is practically useful when it isn't (unless you're a mathematician/theoretical-somethingist seeking to extract tax-payer money)
There's little need to get linear algebra in you unless you want to waste time.
Those people may exist. But I don't see any of them here.
You've gone too far in the other direction by saying it's a stupid and complete waste of time to study math. There are real breakthroughs in insight there, even if sometimes obscured by pompous nonsense.
Your comments are fueled by anger, rather than a sincere effort to inform others, and this is why you've been downvoted.
I'm willing to bet you can be a constructive contributor here if you try. You're probably right that a simplified and more pragmatic approach to learning math might make it accessible to a larger group of people. If so, it would be better to go help make that happen than to blindly criticize everyone else.
There's a huge, huge range of frauds being committed in society. I think people should point them out when they see it.
edit: I also agree mostly with the hard-headed engineers you brought up. Physics is somewhat useful, but it's also vastly overtaught and overfunded relative to it's practical utility. I'm an AI guy and think we'll get the singularity before any of the fundamental research going on pays off (and that's only a small proportion of the world's wealth being spent on physics education for people who will never find practical use for it). And I think there's also a lot of mathemagic symbol throwing in there as well to extract tax-payer money (a lot of it caused by the mathemagicians's influence on physicists).
Also, don't feed the troll.
1. Yes, I mixed in linear and non-linear processes. My point is that you use linear algebra even when working with non-linear processes.
2. Yes, I poo-pooed spreadsheets as a bad way to market linear algebra (I thought that was clear?)
3. Yes, I implied infinite dimensions is useful. Linear algebra in QM tends to be infinite-dimensional.
I'm reading a lot of hostility towards mathematics in general and linear algebra specifically (you've put a bunch of toxic comments all over this thread), and I'm not really sure why.
You don't have to appeal to QM for their usefulness. Function spaces are important in applied and computational mathematics because of their use in understanding integral equations (the theory of integral equations before Hilbert spaces was a giant mess), partial differential equations, calculus of variations, approximation theory, etc. No doubt marshallp will respond that this is all bullshit because it ultimately boils to finite processes running on finite state machines, where there are no infinite sets in sight, let alone infinite dimensions. The equivalent approach to physics would be an extreme form of empiricism, banning the use of concepts like electrons (as some logical positivists actually proposed to do in the early twentieth century) and requiring all physical laws to be stated in terms of directly observable phenomena, whatever that means.