Proving Lagrange’s theorem isn’t too much to ask from someone in a college level math class. We certainly had to in our introductory algebra class our freshman year. That said, it feels a bit out of place in a lin alg class. Being able to prove something like the rank-null space theorem, on the other hand? That should be fair game.

It sounds as if your professor is teaching lin alg the way an honors lin alg class is taught, making sure to generalize to vector spaces over different fields and possibly even modules over arbitrary rings? And the class focuses on proofs and not calculations? There’s nothing wrong with that per se, and personally I think it is much more useful to spend your time in college getting a hang of the deep theoretical results, because calculations are relatively easy if you understand the theory.

If the class is too hard for you as a group, you should bring that up with the professor. Try to be specific about what is difficult and what has been good about the course. Otherwise, don’t be discouraged when the material is hard. It’s when you’re forced to struggle with new concepts that you’re really learning something.

Is that really not normal? I always thought it was like this everywhere and some semesters later I feel like the generalization (eg of vector spaces over different fields) were helpful.

The material that you have described (e.g. Lagrange's Theorem) is perfectly well within the grasp of an undergraduate algebra course. The problem is simply that you are learning abstract algebra instead of linear algebra. These are very different subjects, which serve different purposes for mathematics majors. Since linear algebra is one of the most important courses in your major, you should probably make a complaint to your department's chair, and/or the dean, or relevant higher-ups. Tell them that you signed up for a linear algebra course and instead were taught something else. There should be a list of student learning outcomes somewhere which describes the material that is supposed to be covered. It is a problem if you are not taught the material on that list.

As a side note, I find it odd that someone whose background is in numerical analysis would be pushing a linear algebra course toward topics in group theory and ring theory. Linear algebra is a really huge component in numerical analysis, while abstract algebra is nearly disjoint from that field.

Seems very weird to me that he's teaching ring/group theory at all in an intro linear course, forget how deep he's going...

It's not uncommon to teach linear algebra with abstract algebra vector spaces and fields. Rings and groups in depth don't fit into that course though.

The thing that makes linear Algebra so beautiful is that it can be understood from so many different perspectives. I tend to think about it more algebraiclly so can't help you with the visuals beyond "go watch the 3Blue1Brown video series!!". The more general advice is to see where and how other fields use LA or functional analysis and slowly roll that into your intuition!

For the general student who is 'locked in' to neither 'pure' maths or 'applied' maths (terms I don't like a lot personally, but using provisionally), I would begin from the familiar, i.e. 1. solving systems of linear equations, and 2. transformations, and then move on to the formalisation of vector spaces.

Pedagogically, because linear algebra is often an early topic in (university) maths education, I would take a minor detour here to emphasise why the abstraction is so powerful, because anytime we dive into abstractions that don't seem intuitive, a common question (articulated in words sometimes and sometimes not) is why go to these lengths at all, or whether the formalisms are just a philosophical curiosity (e.g., What's the minimum we need to assume to construct familiar structures?) or 'useful' for some definition of 'useful' (I'd even argue that a philosophical reflection is not useless, but obviously, we mean here uses beyond just intellectual rigour).

If you're looking for resources, I generally recommend Strang (more computational) and Lang (proof-based). Another comment mentions 3Blue1Brown's videos, which are also a great resource for the geometric intuition.

You didn't mention it explicitly but from my experience the most unmotivated construct is the determinant. I finally got it geometrically after i saw how it appeared naturally in exterior algebra, and tried finding the geometrical foundation to clearfy why volume have to be signed/oriented for it to be multilinear.

-6.5 diopter, but that's just because I have poor eyesight

You mentioned that functional analysis made more sense to you. A common approach to a first course on functional analysis, which I think applies to linear algebra very well, is you can think of it as a study of two things:

 

1. The geometry of the spaces you are interested in
2. The linear operators between those spaces

 

In linear algebra, these things also apply but in someways simpler because you work with finite dimensional vector spaces. What’s interesting is, since you are working with finite dimensional things, in some ways, it feels richer since you can actually work and do computations with the things you’re studying.

Also, in math, the larger the breadth of a theory, the less you can say since it needs to apply to more things. Linear algebra being a “smaller” field, has “more” results because in a sense you are working with “fewer” things.

I can understand some of the topics like eigenvalues and determinants but row echelon and vector spaces are difficult for me to understand

David lay, Linear Algebra and Its Applications, this book is the best out there to ace linear Algebra. I did not follow the class's recommended book since I found it to be not quite organized and very ambiguous

Watch Prof. Strang's (from MIT) videos on YouTube. He has a great way of explaining things.

Check out 1Blue3brown's series on YouTube. Just watch the series without taking notes, then review some of the concepts and come back to it again and you might start to connect things

Try to really understand the definitions both intuitively and formally. Obviously solve problems. One of the key problems students usually have is not understanding what one is being asked in a question. It's also crucial to focus on what one needs to prove.

For example consider the following exercise:

Let A be an nxn invertible matrix and B={v1, ..., vn} linearly independent in Rn. Prove that C={Av1, ..., Avn} is linearly independent.

One needs to focus on "Prove that C={Av1, ..., Avn} is linearly independent.".

The one asks oneself what is the definition of linear independence (or maybe theorems involving linear independence). Then one sets up an equation:

a1*Av1 + ... + an*Avn = 0

and one must understand that one need to prove that

a1 = a2 = ... = an = 0

This is an "easy" question. However students have trouble with this because at times they don't understand what they need to prove or don't remember or understand the notion of linear independence.

In addition linear algebra is quite abstract and has loads of definitions.

Vector spaces are an abstractions of Rn that have a "nice structure". In order to define linear transformations one needs to have the notion of a vector space.

Try to get a good book, understand the definitions and solve problems. Also if it's a university class then go to office hours, go to class and try not to fall behind.

As an aside I do have a problem solving course in linear algebra that you are welcome to check out, however your highest priority should be to solve the class problems and sample exams.

Happy Linear Algebra!

EDIT: I forgot to mention 3blue1brown is great for intuition however it's not enough to pass a class or solve problems.

I always siggest the following: take any course material (preferably not too abstract, go for LA for engineers, phycisista, chemists, etc.) - and try to understand everything first in term of simple 2- and 3-dimensional geometry. You will be surprised how much of linear algebra is nothing but a generalization of very intuitive facts about these spaces.

Read Jim Hefferon's Linear Algebra book. Workout the example problems of it. It was awarded the best book for undergraduate linear algebra. There are video lectures of his on YouTube too.

Try to understand the real and complex number systems as a field

Get Otto Bretscher's textbook and read it thoroughly for the geometric intuition. If you want more algebraic calculations and proofs, Insel, Spence, and Friedberg is pretty decent. Linear Algebra Done Right by Sheldon Axler is even more rigorous. Gilbert Strand, Serge Lang, and Johnson, Riess, and Arnold also have textbooks.

Read various textbooks and how they tackle the explanations. Write down theorems, proofs, and examples. Then work out the calculations for simpler computational problems before doing proofs.

Maths concept of field

come dm i've a lot of challenging problems

Prove V* tensor W is canoncally isomorphic to Hom(V,W) with the universal property of the tensor product.

Ladr exercises

Check out two books:

Linear Algebra Done Right, by Axler

Linear Algebra Done Wrong, by Treil

It's beautiful as hell. omg the infinite dimensional stuff. When I first read that, I felt like Voldemort discovering dark magic. A basis of the derivative! Still don't know whether linear algebra gets at the heart of calculus or the other way around.

Don't grind yourself manually inverting matrices. You're not a human calculator.

Im currently refreshing it for my calc II exam in March, it's one of 4 chapters and usually the easier one to get up to speed.

But yeah, solving systems of linear equations, calculating determinants and eigenvectors or solving vector geometry problems does get boring after a while.

For me the fun part is actually using it, like writing code that solve SLEs for you, rotate something in 3d etc. Thats why I find it not too bad, just knowing there is a good chance it might come in handy later on.

So, if you want advice how to make it fun (and if you are into that kind of thing) : try to write some code in an easy language like python that uses some stuff from LA to actually do something.

3blue1brown has a series I love on YouTube called "The Essence of Linear Algebra" that focuses on pictures and geometric interpretations. I recommend it.

Yes! I learned from a professor that got paid by the authors of the book he was teaching from, to in fact teach from that book. The book wasn’t even finished! The book was a ripped off version of another poorly written book, so they made it a “Dynamic Learning” book.

That just means they don’t explain anything and expect you to have a Chegg account. linear algebra isn’t hard, but if it’s poorly taught, it’s nearly impossible to learn.

It has a very beautiful theory that is too often presented just for engineers/scientists who only need it for computation. If you take a 300 level class in it you will get to see the true beauty of it.

Linear algebra books and resources generally come in two flavors: computational and proof based. If it's a proof based problem you're stuck on then I'm not surprised since proofs are generally not covered in HS. I recommend reading the Velleman book to get up to speed on proof.

Take breaks. If you're really stuck, leave it for a bit and do something else (mathematical or otherwise) and come back to it. If you work 100% on one thing you can easily get trapped into thinking about it in one way, so when you get back into it with a refreshed mind you might think of some other approach and new insights. Another thing that can help is to try to do things with a different medium - write things up properly in latex, explain what you're doing to somebody else, or just lie down and think without writing anything down. These will all force you to approach things in a different way and this can be helpful too.

Research progress is like a step function. You spend a lot of time not making progress and then you make big leaps very quickly now and then.

I just try not to rush it and make sure that I keep trying different things even if I'm not making progress: reading papers, working examples, restating the problem, etc.

Usually I talk to people about the idea I’m stuck on. Even if they are not specialised in it, it’s helpful to try and verbalise exactly what problem you are having. And occasionally I’m offered advice that allows me to make decent progress. Otherwise, it can be helpful trying different flavours of resources which all try to explain the same concept and seeing if I can translate between them.

Why not Google it? Since linear is an intro class, you should be able to find it online. Now if you really are trying to learn it, then I find skipping it for a while and moving on. Sometimes when I do other problems or proofs, I see a technique that I didn't think of when doing that problem. I've also figured out showing if something is a vector space by filling in the details of the proofs, reworking the details of the examples in the ebook, or trying to show it is not (proof by contradiction).....

If you need some practice problems now, Jim Hefferon's Linear Algebra is a free textbook available online (as a pdf) on the topic, and it has, like 50ish, problems for each of the chapters, and each problem has a worked solution in the accompanying solutions book. You can also buy the physical copy, if you'd like as well, and there's an accompanying youtube course.

Try any of the books / solution manuals here: https://math.solverer.com/spaces/linear_algebra. All of these problems come with step-by-step solutions so you can always verify your calculations.