- #1
mesa
Gold Member
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There, that feels better...
There, that feels better...
There, that feels better...
It's dry at first, but when you see how it starts to tie all the other math together you've learned - differential equations, orthogonal polynomials, Fourier series, and obviously the stuff you do in QM which you may not have realized with linear algebra the whole time... it's actually a pretty cool subject.
Matrices, however, will always be dry to me I think - unless I'm programming them.
Just curious: did you like Geometry?
Linear algebra can be viewed as the theory of vectors (and vector spaces), and their linear transformations ... which are the matrices when you have chosen an explicit basis.
Much of the rest is the mechanics of how to do this, and the conditions that apply.
There are many important applications, from the Schroedinger equation (H |psi> = E |psi>), which is an eigenvalue equation, etc.
And the student. It takes a certain kind of person who can appreciate abstract mathematics as a thing of beauty. Other people want, or even need, to see that abstract mathematics be made concrete (see how it is applied) before they can begin understanding it. Those other people probably shouldn't be math majors.There seems to be a love hate thing with this subject, and I have a feeling it has a lot to do with the teacher.
And the student. It takes a certain kind of person who can appreciate abstract mathematics as a thing of beauty. Other people want, or even need, to see that abstract mathematics be made concrete (see how it is applied) before they can begin understanding it. Those other people probably shouldn't be math majors.
The same goes with the sciences and engineering. I know I've seen posts here by mathematicians who just don't quite grok science. They understand the math with no problem, but the how and why they should use this math or that is a struggle. They found a much better fit to the way they think over in math world.
I am already convinced of linear algebras usefulness, the problem I am having is we are just learning the basics and no one seems to be able to show how these things work. If on an elementary level it is difficult to 'see' then I fear what is to come.
I have no idea if this is similar to what you are looking for, but it definitely a "pictograph" of least squares regression to "see" what the math is doing.
Linky to the page
It depends. Least squares regression? Absolutely. That least squares is linear is what makes it so easy. On the other hand, a lot of robust estimation techniques are not linear. Many aren't even differentiable (e.g., minimax techniques). Robust estimation is not easy. To make it somewhat tractable, most (all?) robust estimation techniques make locally linear approximations -- and then rinse and repeat.Hmmm. Is linear regression part of linear algebra? Actually it is discussed in the applications section of a textbook of mine, But it also discusses applications of LA to differential equations.
And the student. It takes a certain kind of person who can appreciate abstract mathematics as a thing of beauty. Other people want, or even need, to see that abstract mathematics be made concrete (see how it is applied) before they can begin understanding it. Those other people probably shouldn't be math majors.
I am a firm believer in that we shouldn't use something unless we understand it.
So much for engineering!
but finite dimensional LA is very boring I agree. The spaces are way too well behaved.
Just curious: did you like Geometry?
I love it, favorite subject by far although it seems to be a bit of a lost art today with so much emphasis towards Calculus for geometric derivations (at least for the engineering program at my school). I love what calculus can do but I prefer the old school :)
Linear algebra can be viewed as the theory of vectors (and vector spaces), and their linear transformations ... which are the matrices when you have chosen an explicit basis.
Much of the rest is the mechanics of how to do this, and the conditions that apply.
There are many important applications, from the Schroedinger equation (H |psi> = E |psi>), which is an eigenvalue equation, etc.
That's a bit naive.
...but it's amazingly useful and should be in everyone's toolbox.
I'm a geometry lover myself but there's a lot of bias in there because of general relativity :)
I am a firm believer in that we shouldn't use something unless we understand it. For me this applies to all mathematics, even the basics like being able to derive Pi, e, the quadratic, identities, etc. etc. (and not through memorization but by actual reasoning). If the way of mathematicians today does not follow this path then I am saddened by the state of affairs for such a wonderful subject.
I'm fairly neutral, although I prefer applied math, makes me able to observe the outcome so to speak.
Linear algebra isn't anything horrible, I don't understand what the hate is about - this was also the case during our vector algebra course, some people just DETEST it ..I just..I don't know, I give up..
There are chances that your symptoms of "allergy" to Linear Algebra are due to its strong coordinate-dependency. Indeed, working with matrices as blocks of numbers will inevitably obscure very often the underlying geometry of many operations.
In Geometric Algebra you work more abstractly with elements of a vector space which have a very tangible geometric interpretation, and without directly *representing* them with matrices. This is essentially a modern, and coordinate-free approach to the subject.
You will definitely have to master some new algebraic techniques (i.e. Clifford Algebra), but you won't be mechanically manipulating array of numbers or multiplying big matrices.
However, at some point you will learn that matrices (and tensors) are in fact, a convenient representation of elements (and transformations of elements) of a Geometric algebra.
Warning: if you are currently taking a course in linear algebra, and you have time constraints to study, absorb the concepts and pass the exam, then diving into geometric algebra might confuse your ideas, as translating back and forth between the two languages is not a trivial task for a beginner.
I will certainly look into this. I checked my local library and they have an (electronic) copy of Dorsts book, I look forward to reading it.