- #1
mesa
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There, that feels better...
mesa said:There, that feels better...
mesa said:There, that feels better...
dipole said: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.
lisab said:Just curious: did you like Geometry?
UltrafastPED said: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.dkotschessaa said: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.
D H said: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.
mesa said: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.
nitsuj said: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.
http://hspm.sph.sc.edu/courses/J716/demos/LeastSquares/LeastSquaresDemo.html
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.SW VandeCarr said: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.
D H said: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.
mesa said:I am a firm believer in that we shouldn't use something unless we understand it.
dkotschessaa said:So much for engineering!
WannabeNewton said:but finite dimensional LA is very boring I agree. The spaces are way too well behaved.
lisab said:Just curious: did you like Geometry?
mesa said: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 :)
UltrafastPED said: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.
D H said:That's a bit naive.
jhae2.718 said:...but it's amazingly useful and should be in everyone's toolbox.
WannabeNewton said:I'm a geometry lover myself but there's a lot of bias in there because of general relativity :)
mesa said: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.
lendav_rott said: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..
mnb96 said: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.
mesa said: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.
I have always been passionate about mathematics and science, but I struggled with understanding and appreciating linear algebra. Through my own experiences and research, I realized that many students also struggle with this subject. I wanted to share my journey of overcoming my dislike for linear algebra and provide helpful insights for others who may be struggling.
My journey involved a lot of self-reflection and understanding the importance of linear algebra in various fields of science and technology. I also sought help from professors and peers, and actively engaged in practice and application of the concepts. Through persistence and dedication, I was able to overcome my dislike and develop a deeper appreciation for linear algebra.
One common misconception is that linear algebra is only applicable in mathematics or engineering. In reality, linear algebra has numerous applications in fields such as physics, computer science, economics, and even biology. Another misconception is that linear algebra is just about solving equations, when in fact it involves concepts such as vector spaces, linear transformations, and eigenvalues that have broader implications.
My biggest advice is to not give up and seek help when needed. Linear algebra can be a challenging subject, but with persistence and support from professors and peers, it can be conquered. Also, try to understand the real-world applications of linear algebra and how it connects to other subjects. This can help make the concepts more interesting and easier to grasp.
Linear algebra has numerous applications in scientific research, particularly in fields such as physics, engineering, and computer science. For example, it is used for modeling physical systems, analyzing data in experiments, and developing algorithms for machine learning. Linear algebra also plays a crucial role in solving complex equations and systems of equations, which are often encountered in scientific research.