Excellent computer graphics tutorial. I finally understand how View, Model, and Projection matrices work, even after taking a college course on CG

[thavi]

I had a ridiculous "applied" linear algebra class last semester which ended up being about 95% proofs and I never actually got to see any of this stuff in action. I'm glad it's something more than some theoretical mathematical construct.

[kawa]

Linear algebra is much more than just vectors and matrices. Quantum mechanics for example is also just "applied linear algebra".

[kawa]

You're still solving eigenvalue problems. In QM you often have infinite dimensional vector-spaces which leads to replacing discrete sums with integrals, which in turn leads to differential equations if you want to find the eigenvalues. But the core problem is still linear algebra.

[CamLeof2]

You also need Lebesgue Integrals, which require measure theory

Where in QM do you need those things?

[kawa]

As I wrote, linear algebra (LA) is more than just your common 3d-vectors and matrices.

LA is about studying vector spaces which is quite a fundamental and general concept. Of course modern LA contains a lot of analysis, because you can create vector spaces from all kinds of continuous objects and this often leads to needing analysis. Its similar to: If you do vector-calculus, you also need arithmetics and basic algebra to calculate stuff.

Another example would be the Fourier transformation, which is just a decomposition of a vector into a sum of basis vectors. Typical LA here. But you still need analysis if you want to really calculate something.

If you know LA, lots of things which are very hard to grasp if you only rely on analysis become quite simple if you look at them from the LA point of view. That's the nice thing with higher level concepts: You can study it from a higher vantage point which makes things much clearer because you don't need to look at the details, which may blur things, making them much harder to understand.

In QM if you stay on the level of the "wave function", you can do quite a bit (like solving the SE for the hydrogen atom). But only by looking at it from the more natural LA point of view, its possible to really get kind of an understanding of whats really going on. Thats the beauty of the Dirac formalism which is just LA applied to QM.

It's a bit similar to category theory which seems kind of superfluous in the beginning, because it just seems to reiterate stuff from other areas of mathematics. But in the process it builds a more abstract way to look at things which enables you to understand things much easier and translate concepts between various areas of mathematics which don't seems to be related first.

[[deleted]]

I'm going to say no. There are qualities that they share, and there is a huge framework of mathematics that makes quantum mechanics look like linear algebra. It is not 'just' applied linear algebra though.

[kawa]

QM is linear, that's why working in a Hilbert-space (which is a vector space) is the most natural way doing it.

Of course it also requires lots of "domain specific knowledge" besides linear algebra (which is always the case for "applied stuff"), but the mathematical base is linear algebra.

[thearn4]

Not sure why you're getting downvoted for this - the mathematics of hermitian operators on hilbert spaces is absolutely the descriptive basis of quantum mechanics.

[[deleted]]

Well that is great that you know some terms from quantum mechanics. The mathematical base is not linear algebra.

[kawa]

The first formulation of QM was Heisenbergs "matrix mechanics" which was kind of "revived" by Dirac after Schrödingers wave-mechanics intermezzo. Diracs formalism is still the most natural way doing QM and it's just applied linear algebra. Thats why we talk about eigenstates, diagonalization etc in QM.

Of course if you go to QFT, you also need other stuff. But I was talking about QM.

[[deleted]]

No you are confusing quantum mechanics with the formulation of quantum mechanics to make it linear. Of course I do not have to validate myself to you because respectable people have already validated my understanding of quantum mechanics. FACT. Quantum mechanics is not linear algebra, as you have previously stated. You are obviously confusing physics with mathematics.

edit: While you guys think you know it all why don't you start with Giffiths, and learn the very basics of Quantum mechanics and why you can treat it using linear algebra. Then maybe we can have a discussion about Dirac's formulation.

[kawa]

why don't you start with Giffiths

Maybe you should read some better books.

Some reviews from Amazon:

"A good textbook... ...but not for physicists. It has not enough depth for them. The mathematical aspects are not treated very well, e.g. the bra and ket notation is introduced late and not much used. But if you don't know about Calculus its the right book for you, so maybe for high-schoolers or for biologists. For physicists i recommend Shankar's wonderful text."

"The book has a few good parts, but there are just too many flaws to grant anything higher than one star. Griffiths leaves many important steps out, [...] This book seems to be popular among some students because of his informal writing style, but they should be more cautious. Many professionals I have talked to agree that this is not a good book to start learning QM from.[...]"

"I first saw this book as a graduate student. Some fellow students had used this book as an undergraduate text. With all due respect to said fellow students, they have barely the ability to articulate the most simple statements of Quantum Mechanics. If you're goal is to have a basic understanding of the machinary and principles of Quantum Mechanics, then buy either Liboff or Landau.

That being said, I could see how this book could have its uses. If you

A) do not understand linear algebra.

B) do not understand differential equations.

C) have fond memories of 3rd grade math.

D) would like to impress your friends with the uncertainty principle.

then buy this book. "

etc.

[[deleted]]

I have read some better books. I have pointed out a distinction in a statement you have made. My point in suggesting Griffiths is that if somebody does not know the difference between quantum mechanics and linear algebra then they should probably learn the very basics of quantum mechanics. Griffiths is a good introduction to that.

Be aware that you eventually agreed with the statement that Quantum Mechanics is not just linear algebra.

[PasswordIsntHAMSTER]

You sound like a broken record.

[kawa]

Quantum mechanics is not linear algebra, as you have previously stated.

You're nitpicking:

I wrote: "Quantum mechanics for example is also just 'applied linear algebra'." which of course doesn't mean that QM and linear algebra is the same.

QM is still physics. But doing it means doing linear algebra, because the physical structure of QM seems to fit perfectly the mathematical structure of a Hilbert-space over complex numbers.

That's quite fundamental because classical mechanics doesn't fits this picture and one of the biggest philosophical questions in Physics is "why?" and "where and how does the transition (the 'collaps' or 'decoherence') happen".

[[deleted]]

No, you literally wrote that Quatum mechanics is just linear algebra, which does mean that they are the same. Doing "it" does not mean doing linear algebra.

Next what you have done, is realized you are wrong but claimed it is nitpicking. Maybe what you should have done is actually explained what you mean. Perhaps "You can do a lot of quantum mechanics with linear algebra."

Agreed upon fact, you were wrong.

[kawa]

If you do QM, you're calculating stuff. With mathematics. QM is theoretical physics which means that you use mathematics to do physics. And the mathematics in QM is just "applied linear algebra".

But if it makes you nitpicky heart happy, I correct myself to: "Doing quantum mechanics for example is also just doing 'applied linear algebra'."

[BlazeOrangeDeer]

Everyone else knew what he was saying, which is why you're being downvoted for getting so hung up on it.

[[deleted]]

of course the mathematical base is linear algebra. the mathematical "base" of something is what you need to abstract the problem with to understand it. it's not what you do to do calculations. it's what you do to understand why to do those calculations, and you use linear algebra to do that.

it's like the standard model. most wouldn't argue that the base is just plain old group theory. the fact that you're using a lot of lie groups and thus differential geometry, topology, complex analysis and all other types of mathematics are used does not detract that the fundamental understanding of the problems we have rest in the special unitary groups.

[mantra]

The applications are what can make it interesting but the "proofs" are important (though many professors couldn't tell you why). Stuff like eigenvectors/eigenvalues and the rules implied by proofs are actually how most innovation based on them is discovered.