# I hate linear algebra!

Gold Member
There, that feels better...

## Answers and Replies

inotyce
There, that feels better...

dipole
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.

Staff Emeritus
Gold Member
There, that feels better...

Just curious: did you like Geometry?

Gold Member
IMHO, you can never know too much linear algebra!

Gold Member
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.

From what I have seen there is no doubt about it's usefulness, its the not being able to see what I'm doing that is frustrating :P

I picked up several supplementary texts but they all seem to emphasize proofs of the procedures as opposed to why and how they work. Even Anton's book on the subject was dry, yikes!

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 :)

Gold Member
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.

ZeroPivot
what about non-linear algebra =)

Gold Member
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.

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.

On the subject of vectors, no doubt trigonometry combined Descartes wonderful coordinates is a powerful combination of techniques by bringing in coordinates to trigonometry. I am routinely amazed at it's usefulness and the insight it provides as this is a whole new way of doing trig for me.

Going just outside the realm of the textbook (Calc III) we can see how vectors can be used for solving certain types of second degree multivariable polynomials, WoW! This is one subject area I will be dedicating much time to during winter break!

It is a shame there isn't more time for exploration during the semester. In the meantime I hold out hope there will be a reasonable tie in with linear algebra at some point...

dkotschessaa
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. It's actually a very beautiful subject. The way some of the proofs and identities and things work out - it's very nice and neat. Not messy like calculus. Nice and..well, linear. I don't know how else to say it. don't give up!

Staff Emeritus
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.

dkotschessaa
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.

Yes, true. I ultimately found that i had a love for abstraction, and that applied mathematics was tedious and messy. All those natural forces of the universe kept getting in the way of my pretty equations. so I became a math major.

However, I know a lot of math majors (and at least one teacher who has a masters) who are abstraction-oriented but for some reason still do not like linear algebra. Maybe it's still too 'practical' for them?

-Dave K

nitsuj
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

SW VandeCarr
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

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.

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Staff Emeritus
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.
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.

Gold Member
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 was unaware that abstract math immediately equated to techniques that very few people understand. These processes did not just 'appear' on a sheet of paper, they were made by people who developed them. In this class (and the books I have seen so far) it seems the emphasis is primarily on the technique while the way how these ideas came to fruition are forgotten.

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.

dkotschessaa
I am a firm believer in that we shouldn't use something unless we understand it.

So much for engineering!

Gold Member
So much for engineering!

Hah! :D

Staff Emeritus
That's a bit naive.

You haven't told us the book, and you haven't even told us the target audience of the class. Is this
• A linear algebra class for math majors,
• A linear algebra course for non-math majors, or
• An applied mathematics class that covers linear algebra along with other stuff (and just when you think you are starting to get one subject the instructor will say "and now for something completely different", switching to integral equations and Green's functions)?

LA is extremely dull until you get to infinite dimensional topological vector spaces. Then stuff gets interesting :) but finite dimensional LA is very boring I agree. The spaces are way too well behaved.

Homework Helper
but finite dimensional LA is very boring I agree. The spaces are way too well behaved.

Maybe they are boring at a conceptual level, but figuring out how to compute stuff efficiently (i.e. fast enough so you can still remember what the question was when you have got the answer, and preferably with more than zero correct significant figures in the answer as well) gets a bit more interesting when the space is say 1,000,000-dimensional rather than 2d or 3d.

Staff Emeritus
Gold Member
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 :)

I'm surprised! I absolutely loved geometry too, and linear algebra scratched that same itch.

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.

Yes, vector spaces and mapping transformations are so visual, that's why I loved it. Being comfortable with LA made quantum feel all warm and cozy .

Gold Member
I'm one of those that really needs a physical example in math's and can't think terribly abstractly until after I've seen it in physics. I didn't care much for linear algebra either, but then once I started doing linear stability analysis in dynamical systems, I grew to appreciate it (just a little bit, though, I wouldn't share my ice cream with it).

Didn't appreciate it when I was applying it in QM, because QM, like math, requires abstract thinking. It's really neat using brah-ket notation, though, I guess.

1 person
Gold Member
That's a bit naive.

Well, excuse me then. This is the general discussion forum and I wasn't really looking for more than a quick vent but since the conversation started rolling I decided to step back in.

Now by your post I am pleasantly surprised to hear I am wrong about linear algebra and I look forward to your interpretation of answers I have been unable to find through my class mates, tutoring center, instructors, etc., etc. I'll send a PM when I post.

Gold Member
I do dynamics and controls for aerospace vehicles, so I :!!) linear algebra.

From a pure mathematics standpoint it isn't terribly interesting, but it's amazingly useful and should be in everyone's toolbox.

Gold Member
...but it's amazingly useful and should be in everyone's toolbox.

From the little I have seen I already agree.

It seems many PFer's have a good 'visual' understanding of linear algebra which means there is a critical piece of information I must be missing. If it is anything like geometry (as lisab suggests) then I'll be in good company.

I'm a geometry lover myself but there's a lot of bias in there because of general relativity :)

Gold Member
I'm a geometry lover myself but there's a lot of bias in there because of general relativity :)

Special relativity is wonderful and now I look forward to general relativity. With the new tools from this semester it will be fun to poke around. It must be quite a treat to know it well.

nitsuj
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.

Please become a math teacher lol

Not so sure about the if you don't understand it don't use it when it comes to equations, but the rest is bang on with my experience.

1 person
lendav_rott
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..

Gold Member
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..

From the little I have seen I am already impressed by what it can do but using systems where I can't see how they operate is frustrating, memorizing is a terrible way to approach a new subject. I will try my luck with some of our PFer's and hopefully get this resolved.

mnb96
mesa,

did you try to have a look at some introduction to Geometric Algebra?
If you are an engineer/computer scientist, you might want to have a look at the first chapters of Dorst's book "Geometric Algebra for Computer Science".

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.

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1 person
Gold Member
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.

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.

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.

Then thank goodness for winter break :)

mnb96
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.

Good. Feel free to come back here and tell us what are your first impressions with Geometric Algebra.
In the meanwhile you can have a look at this paper for an informal and historical introduction to the subject.

Anyways, if you are an undergraduate student, I suggest you to master the classical techniques of linear algebra that you will be taught during the course (they are very useful and you simply can't ignore them). However keep in mind that all the manipulations with matrices that you saw (and you'll see) are just a coordinate-dependent representation of geometric concepts.