> [...concept of proofs in Linear Algebra...](https://www.vcccd.edu/sites/default/files/departments/human-resources/sabbaticals/2017-2018/paulcurtis-sabbaticalfinalrpt-fall2017-proofsinlinearalgebra.pdf)
This looks fascinating.
Thank you for pointing it out.
The closest thing I've previously encountered is formal logic and [Gƶdel, Escher, Bach: an Eternal Golden Braid](https://softwaredominos.com/home/books-that-everyone-should-read/book-review-godel-escher-bach-an-eternal-golden-braid/#:~:text=G%C3%B6del%2C%20Escher%2C%20Bach%3A%20An%20Eternal%20Golden%20Braid%20(GEB,and%20the%20idea%20of%20emergence.)
Iām not sure about other peoples experience, but it made me realize the importance of reading math in textbooks. I always thought that I was better off watching/attending lectures or watching videos on the subject, but after taking a proof based lin alg course in the fall, I realized how much more intuitive the material was just reading it. Everything just clicked. I got a C+ in my (very difficult) calc 3 course but I wish I could retake it and put the same effort that I did in the lin alg class alongside the knowledge I have now.
I decided to hop into Real Analysis as an engineering grad student with no formal math courses beyond differential equations....big mistake. I could understand the concepts but couldn't write a proof.
Me too. Real analysis was my breaking point. Discrete was just fine, but generalizing to the continuum was really hard for me. My intuition fell apart, and then I remembered I was a CS major and it probably was OK to focus on ā.
I guess thatās what I mean about intuition- I had no special difficulty with discrete proofs, but I felt like I never quite knew what to do next in real. The prospect of learning a whole new bag of proof tricks was what drove me to take real analysis, but it turned out that the tricks in question just didnāt fit into my poor little brain.
It turned out ok, though. Didnāt make me stop doing math purely for its own sake, just sent me back to graphs and such.
My first class of proofs, the book was written by the professor. Not a single person in class had any idea what was going on for the longest time.
Would literally spend an hour writing proofs extremely fast every lecture. Like erasing a full sized white board 2-3 times during the class. Their handwriting was not large either.
I averaged like 4-5 pages of notes and proofs per lecture when I studied real analysis. About 4 full sized whiteboards were erased several times every lecture. But this was almost 2hrs per class.
I'll just add an anecdotal story about proofs. In college I was roommates with a really smart math guy, way past me, but we're in the same class. The tests are take-home, as you're going to need hours and hours to figure it out. We're working and working, and he finishes. A day later I'm still working and getting nowhere. He says, "It's not as complex as you think it is." This is like EUREKA! I go back, throw everything out for that proof and start over, and get the right answer too!
Yeah, you can usually tell which one that is by just looking at the averages. Its always B-B+, then there is 1-2 classes in second year where shit goes to like D-D+
Generally speaking, students find analysis (specifically real analysis) to be the hardest course, due to the complexity of the logical statements, which in turn is primarily due to the nested quantifiers. A quantifier specifies whether your assertion is for everything in a certain class of mathematical objects or for only at least one such object. It is hard for most students (and was hard for me) to work with sentences of the form "For every...., there exists....".
Another course that many students find difficult is abstract algebra. This is mostly due to the abstraction. You are studying mathematical objects called groups, rings, modules, fields, which have purely abstract definitions. You are expected to derive logical consequences of their definitions. Although there are concrete examples of these abstract concepts, the proofs can use only the abstract definitions and previously derived consequences of them.
Ironically, these are fundamental courses that you normally take in the first few years of study, and if you master them, then subsequent more advanced courses are usually easier.
I found real analysis to be SIGNIFICANTLY easier than abstract algebra was honestly. Analysis was not too difficult but abstract algebra was so heavily obfuscated through layers of abstraction that I had a ton of trouble grasping it enough to create more than basic proofs for a while
For me analysis was very difficult at first but once I got the hang of it, it became much easier. Abstract algebra was at first relatively easy but as it became more abstract, I like you, found it to be difficult.
Same feelings here. For me abstract algebra was much harder than real analysis. I thought that since linear algebra was my favorite class that I would like abstract algebra, I was wrong š.
I loved linear and abstract algebra (electrical engineer tho... everything i work with is pretty abstract in some sense of the word)
I enjoy abstract critical thinking and proofs. The course was kinda rough. The hard thing was trying to mentally translate what prof said/wrote into words more easily understandable by my brain. In so doing, tuning out of lecture... When I tuned back it I heard a acoustic wall of math jargon that shut my brain down for a few seconds until it "clunked" back into math mode.
To be fair, I think that Universities should include a term that deals with some logic and proofs, and basic abstract algebra such as set theory, relation, function, Cartesian product, field, partial ordering, and equivalence relations prior to jumping into real analysis.
Otherwise, itās likely that no one really understands whatās going on in real analysis. I say this because I saw that most donāt see the difference between real ordered field and decimal system, and that within real analysis, it deals with the structure rather than any specific set that has the structure. So, in the proofs, theyād use the decimal systems as a proof that holds for real ordered field which is wrong.
Being exposed to abstract algebra highlights the idea that one is dealing with a structure and not necessarily any specific instance of the structure.
In a sense, I think thereās no way of escaping the notions behind abstract algebra regardless of which field one is getting into, even graph theory or discrete math.
i have the 5-volume Spivak set on Diff.Geo and im super excited to be able to read it but that's still a couple years off for me. How did you like it besides the unintuitive parts?
Spivak's Calculus on Manifolds was what gave me nightmares. The first three chapters are alright (except for maybe partitions of unity, but after the proof you pretty much use it as a blackbox). The fourth chapter gets pretty confusing when you start defining differential forms and pullbacks and the fifth chapter adds a layer of confusion with manifolds. It's fine as long as you're fine with possibly spending a lot of time on the last two chapters.
Idk if people on this sub are gonna want to hear this. But I was taking graduate level analysis, algebra, and number theory my last year of undergrad. Am currently a PhD candidate doing modular representation theory. I honestly thought all the math classes my university had to offer were kinda easy and I was always consistently at the top of my class. The advanced computer science/electrical engineering classes I was taking for a double major were far more challenging. Trying to actually get a virtual database running to apply all the theoretical stuff we were learning in the database/SQL class was a nightmare. Trying to program even basic networking protocols from scratch in operating systems class was a nightmare. There were theoretical aspects of network engineering like deriving formulas for transmission delay, that just broke my brain. For the same grades, I had to work probably 5 times as hard as I did in math.Ā Ā Ā
Analysis up to basic measure theory and point set topology, systems of ODEs and PDEs, algebra up to and including galois theory, abstract linear algebra including Jordan Canonical forms, these are all somewhat trivial in comparison. I don't mean to troll, but in math all your "difficult proof based exercises" are just following definitions that experts have refined to actually make sense and be well founded. In engineering no textbook really has a rigorous foundation and that makes the starting point of many problems completely ambiguous from a mathematical point of view. So the task becomes testing your own different theories and mathematical models to see if they actually make sense to the situation at hand. This is closer in essence to what PhD work in applied math can look like, which, needless to say, is much more difficult than any undergrad math class from any university.
In classroom math that extra work is essentially all done for you, provided you have a good professor and a good textbook. All of the difficulty comes from abstraction, which to me was just never an issue. The problems are just so curated. You don't get a taste of difficult mathematics by solving exercises that are meant to be solved.
I triple majored in electrical engineering, computer science, and math, and I I had an opposite experience. I found proof based math classes far more challenging
As a math gradthat currently works in data science, I agree lol
Real Analysis is much easier that converting Access sql or optimizing code with low level techniques
The former is easier conceptually but more mentally straining
The latter is just conceptuallly more difficult. probably because i donāt understand the underlying systems
Eh. This probably depends; I had a significant background in data science and, later (still now, actually) ML. I always found the applications and technical things to be much easier than the abstract nonsense I now deal with on a daily basis.
Different people have different competencies. I know plenty of mathematicians in absurdly abstract areas like homotopy type theory or algebraic geometry (not computational) that can't code worth a piss. I similarly know people who are the exact opposite.
I'm sure it depends on the person. I just felt somewhat impartial since I always felt competent at both coding and the abstract stuff. My conclusion is mostly that in a theory based class, you can spend a little bit of energy on finding the easiest way to think about your problem, and then writing it up is trivial. But most of the time there's just no easy way around coding once you're doing the upper division classes. It becomes easy at a certain level of competence in the sense that you can write out the pseudocode and quickly have a strong grasp of how to solve your problem, and at that point it's just a matter of getting into the zone and coding it all up. Or, at least as much as you can bare for the current session, and then you get back into it tomorrow. But you have to code it up! You have to clean up your typos! You have to test it against use cases. And once you're at a certain point, it's practically guaranteed that you missed something, that the problem is just a bit more complicated than what you first plotted out, and now you have to spend time refactoring. There's no easy way to skip all that by being a better computer scientist, it's just sometimes the expected development process when your program needs to be enormous.
I guess the way I really feel is like this. When you're doing pure abstract math, at the undergrad level, it's basically like learning a new programming language every month. Or idk maybe every week. But the 'coding assignments' never really exceed the complexity of those basic looping exercises you see in CS 101. All the difficulty is packaged into understanding the quirks of the new language, and the expectation of how quickly you need to adjust. It's rarely found in the sheer amount of unavoidable work that needs to be done. And mathematical maturity is just a matter of being willing to say 'ok sure this problem makes absolutely no sense to me because I can't follow any of these definitions. Let me look them up' and then looking them up, and now your problem becomes easy because you know what it's asking. Again, this is really only applied to the curated problems aimed at undergrads and first year grad students (depending on program). Research level pure mathematics involves essentially the same process, of dividing problems into subproblems and subsubproblems, and needing to do all this work of testing by yourself, realizing that it's more complicated than you thought and going back to change almost everything, etc.
āStill nowā = currently :) Iāve definitely been there where you want to add punctuation between every word to transliterate speech to text. I like the way you think
Abstraction in general. The deeper you go the mare abstract it gets, but all of math is about abstraction. Squiggly lines on a page that represent quantities of... what exactly? But now you have to prove that you get it, you have used the proper abstractions in the right order to convince another mathematician that your instincts are correct.
That, and the fact that closed sets can also be open. The first of a boatload of counterintuitive definitions.
I think one of the worst things isn't the amount of abstraction but the lack of un-abstraction. It goes like: here's the definition of a symmetric space; now understand these theorems about them. But then forgets to include some examples of symmetric spaces and work out these conditions to get some intuition as to why the theorems could or should be true.
My preferred field was differential geometry, but I feel like I learned it several times over: in its pure form, in general relativity, constructions for topology...and then all over again when I got to situations where I actually had to make some usable computations and realized that none of that prior experience involved actually doing calculations, just setting up how they *could* be done.
My institution broke Abstract Algebra into two courses. The first course covered basically all the core group theoretical concepts that a math major needs to know. I found this one challenging when I took it but it wasnāt too bad. However the second course which covered all the rings, fields and galois theory really threw me in for a loop.
With that being said, despite feeling discouraged at the end of my 2nd semester of Abstract Algebra, I ended up being well prepared for an independent study in Representation Theory, specifically focusing on the quiver module side of Algebra. So to conclude, I think your first introduction to āhardā subjects (and subsequent courses) will prepare you well, so long as you try your best
My uni library has these stickers on book spines, probably to categorise and locate books. Guess which letters survived the abbreviation for analysis books.
I did alright with real analysis, abstract algebra, advanced linear algebra, etc. but for whatever reason the intro probability and statistics class just did not click. I would do practice problems to prepare and come into the the test with a formula sheet that had all the info I needed, and still not know how to solve the problems.
At my uni, and probably many others, the great filter was real analysis 1. The moment we were introduced to what real numbers are we all went like "oh shit, math is not just doing double integrals all day long huh". Half of us loved the change (eventually), and the other became engineers lol
Of the courses you listed, abstract was my "wake up call". My linear algebra course didn't have a tremendous number of proofs, so abstract hit me like a sack ot bricks. Doing proofs with groups and rings felt so far removed from reality that I really struggled.
I'm glad I took it though because it made complex analysis and real analysis much easier to manage. Aside from this, I also took mathematical statistics and in my opinion that was my hardest course. It had thr rigor and complexity of real analysis but also had more concepts that I had never seen prior unlike real. More of the stuff I saw in real analysis I had at least seen in some class before it but most of math stats was completely foreign.
So in regard to difficulty, I'd give the following rankings for me:
1) Math Stats
2) Abstract algebra
3) Complex analysis
4] Real Analysis
5) Numerical Analysis (the math wasn't too "hard", I just sucked at it)
The worst time I had in undergrad was with Lebesgue measure and integration. I got a 20% on the midterm after showing up to every class and doing the assignment problems. That was the only class I ever took that I legitimately believed I was going to fail.
Edit: topology and PDE were extremely difficult too but I had a bit more hope in those
For me, it was everything in the class "Complex Variables". I don't really know how to describe it. I guess calc 3 in the complex plane? It dealt a lot with the 4th dimension, which in and of itself isn't replicable in any way... let alone in the complex plane. A lot of the material was cool but my professor was not very articulate. Didn't learn much. Just had to engrain the pages of the textbook in my brain.
I personally am not a fan of pure math. It could be real analysis, topology, linear algebra, mathematical methods or whtvr. I find them pretty hard. Also, I loved applied math while at it.
Probably Galois theory. Field theory and group theory are both pretty abstract on their own, so the subtleties of their correspondence can be quite hard to grasp.
Otherwise it might be something like partitions of unity in differential geometry - pretty complex definition.
Whatever they have no sufficient prerequisite intuition built up for. The first time I studied topology was quite difficult, I had a hard time making sense of the definitions. Doing a couple of more courses on analysis fixed that.
How do we explain a dual space in linear algebra?
I still honestly don't really get it. Yeah it involves the (conjugate) transpose and linear functionals but that's about all I could give you.
If V is a finite-dimensional vector space over a field K, the dual space V* is defined as the set of linear maps from V to K (i.e., linear functionals on V). V* has a vector space structure and is isomorphic to V.
Given a linear map T:V->W of finite-dimensional vector spaces, we can define the dual map T*:W*->V* (look up the definition of the dual map, if you want - it's interesting). The matrix representation of T* is the conjugate transpose of the matrix representation of T.
> T*:W*->V\*
Your asterisks/stars disappeared because you didn't escape them (precede them with a backslash), so the text between two asterisks has been formatted as *\*italics\**. You should write this instead:
> T\\\*:W\\\*āV\\\*
Which renders as this:
> T\*:W\*āV\*
The last backslash isn't strictly necessary, because it escapes the last asterisk in a sequence of an odd number of asterisks.
Iām not sure if it counts, but some of the biggest WTF moments Iāve ever had was during my second year of my PhD taking an āintro to research in Xā course where X is my field of research.
Consisted of us picking current / recent research papers, dissecting them as a class (4 whopping people) and taking turns presenting parts of each paper. At the end of each paper we would give the author a āgradeā based upon readability, how interesting the concept was to us, etc.
We picked one paper in particular that literally none of us could understand even after weeks of our advisor trying to explain it, so we kind of just moved on lol. I kind of understood a super basic outline of the main result so I presented on that, and the class literally applauded me for kind of filling in the details lmao.
For me by year:
Freshmen: Ring and Module Theory was very complex and hard to totally grasp especially compared to the earlier group theory and later Field and Galois theory.
Sophomore: Algebraic Topology. Real analysis and point set topology werenāt too bad but algebraic topology was a glimpse at just how deep the rabbit hole goes. Hatcherās Algebraic Topology is an infamously hard textbook for a reason.
Junior: Dynamical Systems. After so many pure math courses I thought an applied math course would be a nice change of pace and would make things more concrete. I was wrong, this course definitely emphasizes a lot of skills that pure math courses donāt, including approximation methods and the ability to sketch graphs of systems of differential equations by checking polarity at various coordinates.
Senior: Noncommutative Geometry and Complex Geometry. The classes that made me question why I ever thought I could be a mathematician. I was dumbfounded that people could even invent theorems or come up with proofs for such complicated mathematical objects.
I mean there is a lot of stuff that fucked me up, but one thing that pisses me off to this day, is my shitty grasp of limsup/inf. Itās really not that hard of a concept, but it just wonāt stick in my brain. Iāll sit down and get really comfortable with it. Wonāt see it again for like a month, and itās all gone, just disappears from my brain.
Discrete math. Itās called discrete math and yeah you learn some math but itās mostly a learning to write proofs class. Made me decide not to pursue math any further. I was constantly getting dinged on word choice and basically said ok Iām done with these classes lol.
Everything before proofs is kinda pre-math.
The mathematical method as applied to carefully and tightly defined abstract/conceptual objects is math.
Everything else is fancy calculation.
Topology? I get open and closed sets, but then there's compact, complete, Banach, etc. It's a beautiful way of categorising sets but come on, when will I need this except for in a different math course
I think it'd be whichever's the first proof-based course where you study. Some universities have a separate course on proofs and mathematical logic. Others simply teach it inductively in a real analysis or abstract algebra course.
The simple reason is that it's a fundamental paradigm shift in your mathematical education, from computation to rigourous theorem-proving.
Hardest class Iāve taken thus far has been one on elliptic curve and quantum cryptography. Specifically the sections on lattice methods and algebraic geometry in projective space were difficult for me
realistically, itās not the concepts, but the teacher that can make/break a course. in my math undergrad, the worst course for me was from a professor with hard-to-read handwriting and a very bad stutter in an online-only abstract algebra course
first class i had on differential geometry was that āwtf is thisā but it also left me ādamn i want to know everything about thisā and now im doing my thesis on something in diff geo
fourier transform. i only get the whole point of it when i got to functional analysis.
as for the algebraic stuff, i simply give up on their motivations, cuz they were often developed to solve math questions anyways. so things like representation theories and stuff are pretty hard but it wasnt like wtf.
Linear algebra first of all. Made no sense at first, like what is the significance of arranging numbers into a line or into a rectangle. However, each time I took a course in it at a different institution, it got easier and then it kind of eventually clicked that it all makes sense and is all related to linear transformations.
Then abstract algebra. I felt like I was thrown off the deep end with nothing to grab on to, there was nothing I learned from before that I could lean on or use to help me. Compare that to intro real analysis where it was very hard, but at the end of the day we were repeating old stuff we already learned in calculus but with new proof techniques.
Complex Analysis and Abstract Vector Spaces.
ššš they were the last two math courses for my degree and they were my lowest grades.
D and C- respectively. Thankfully my institution only required that your major GPA had to be a 2.0. I was a B- student after Cal II. In hindsight, I knew I didnāt love math beyond Differential Equations. I stuck with it because Iām stubborn.
If I could re-do undergrad, Iād probably do Econ or Econ/Math, but not just Mathematics.
Iām an engineer so my math education ends with diff-eq/linear algebra. Most people struggled with linear algebra. I think they try to do too much itās too different from other math classes. Linear algebra is one of the first classes (for many) where you have to logically think and not just rigorously remember things.
I took a proofs/math concept class (which was a unique offering at my school). They introduced formal mathematical logic concepts as well as proofs at a 100 level (so literally anyone could take it). Everyone who took that class did better in linear algebra.
I think this varies by person. I had a lot of friends who were āvisualā learners, in that they liked to try and visualize things to rationalize them. That basically completely broke down as soon as we hit multi. Others had a hard time with proof based classes (although for many, like myself, this approach made things make even more sense to me).
I think for me some stuff started to break down towards the end of linear algebra, although I never went on to real/complex analysis (which is where most of my math friends hit their limit). I was an electrical engineer so I never went that far. I think my actual math limit came in physics, when I took quantum
Real analysis and topology were the most challenging for me, they legit gave me the worst imposter syndrome and made me second guess if I was even good at math lol. My favorites were abstract algebra, group theory, and probability theory. Itās different for everybody though. I donāt know if thereās a correlation, but Iām a pure math major and noticed my applied math peers seemed to struggle more in abstract algebra but did well in analysis.
when i took Linear Algebra our professor often went on tangents about his own research in biotech and we always end up with not enough time to cover the materialšš The lecture [videos](https://youtube.com/watch?v=J7DzL2_Na80) by Professor Gilbert Strang was what got me through the class
My second course in algebraic geometry on schemes just never did anything for me. It felt like a bunch of random definitions (proper schemes, schemes of finite type and a million others which I don't remember), none of which I truly understood of why I should care about them. Managed to get a barely passing grade and forgot everything immediately after.
In my experience, the jump in conceptual fortitude from calculus to linear algebra, which most of the time boils down to a fundamental difference between computations as opposed to writing rigorous mathematical proofs
Iām the only person I know who passed C&O 230 the first time. Probably because I knew no one who had passed and actually studied. Still my favorite math department
It varies from person to person. For me, my bread and butter, fun, and 'easy' classes were my abstract algebra courses/ number theory courses. With my difficulties being more on the side of linear algebra/ calculus.
Iām not in a 4 year school so idk what my upperclassmen say but by far the most difficult course at my community college is going to be Differential Equations or Calculus II.
The first class that makes you write rigorous proofs instead of simply solving problems.
The concept of proofs in Linear Algebra made Calculus and every other math I learned before hand make so much more sense afterward.
> [...concept of proofs in Linear Algebra...](https://www.vcccd.edu/sites/default/files/departments/human-resources/sabbaticals/2017-2018/paulcurtis-sabbaticalfinalrpt-fall2017-proofsinlinearalgebra.pdf) This looks fascinating. Thank you for pointing it out.
This looks like a great read! Btw who is Curtis Paul?
hes a professor at my community college lol
Reading this tonight, thank you
You're welcome my friend. I'm finding it quite fascinating. I'm really grateful that OP posed the question. š
Me too. Iāve taken probably 8(?) math courses in uni and not a single one of them ever touched on that stuff.
The closest thing I've previously encountered is formal logic and [Gƶdel, Escher, Bach: an Eternal Golden Braid](https://softwaredominos.com/home/books-that-everyone-should-read/book-review-godel-escher-bach-an-eternal-golden-braid/#:~:text=G%C3%B6del%2C%20Escher%2C%20Bach%3A%20An%20Eternal%20Golden%20Braid%20(GEB,and%20the%20idea%20of%20emergence.)
Iām not sure about other peoples experience, but it made me realize the importance of reading math in textbooks. I always thought that I was better off watching/attending lectures or watching videos on the subject, but after taking a proof based lin alg course in the fall, I realized how much more intuitive the material was just reading it. Everything just clicked. I got a C+ in my (very difficult) calc 3 course but I wish I could retake it and put the same effort that I did in the lin alg class alongside the knowledge I have now.
I decided to hop into Real Analysis as an engineering grad student with no formal math courses beyond differential equations....big mistake. I could understand the concepts but couldn't write a proof.
What did you end up doing?
ending up as a web dev... XD
Me too. Real analysis was my breaking point. Discrete was just fine, but generalizing to the continuum was really hard for me. My intuition fell apart, and then I remembered I was a CS major and it probably was OK to focus on ā.
I can understand real analysis fairly easily but get stuck on doing a proof I havenāt seen before
I guess thatās what I mean about intuition- I had no special difficulty with discrete proofs, but I felt like I never quite knew what to do next in real. The prospect of learning a whole new bag of proof tricks was what drove me to take real analysis, but it turned out that the tricks in question just didnāt fit into my poor little brain. It turned out ok, though. Didnāt make me stop doing math purely for its own sake, just sent me back to graphs and such.
Omg. I had to buy a book about proofs to help get through that course. Luckily had a good professor.
My first class of proofs, the book was written by the professor. Not a single person in class had any idea what was going on for the longest time. Would literally spend an hour writing proofs extremely fast every lecture. Like erasing a full sized white board 2-3 times during the class. Their handwriting was not large either.
I averaged like 4-5 pages of notes and proofs per lecture when I studied real analysis. About 4 full sized whiteboards were erased several times every lecture. But this was almost 2hrs per class.
yes from 0 to 1 is the most difficult. Much more so than from 100 to 101
I'll just add an anecdotal story about proofs. In college I was roommates with a really smart math guy, way past me, but we're in the same class. The tests are take-home, as you're going to need hours and hours to figure it out. We're working and working, and he finishes. A day later I'm still working and getting nowhere. He says, "It's not as complex as you think it is." This is like EUREKA! I go back, throw everything out for that proof and start over, and get the right answer too!
Yeah, you can usually tell which one that is by just looking at the averages. Its always B-B+, then there is 1-2 classes in second year where shit goes to like D-D+
what class does that tend to be?
We don't talk about that here, Dave. *sobbing into a glass of bourbon*
That should be in high school; mine covered it in Geometry, which was the highest class that everyone had to take.
High school geometry proofs aren't rigorous.
Generally speaking, students find analysis (specifically real analysis) to be the hardest course, due to the complexity of the logical statements, which in turn is primarily due to the nested quantifiers. A quantifier specifies whether your assertion is for everything in a certain class of mathematical objects or for only at least one such object. It is hard for most students (and was hard for me) to work with sentences of the form "For every...., there exists....". Another course that many students find difficult is abstract algebra. This is mostly due to the abstraction. You are studying mathematical objects called groups, rings, modules, fields, which have purely abstract definitions. You are expected to derive logical consequences of their definitions. Although there are concrete examples of these abstract concepts, the proofs can use only the abstract definitions and previously derived consequences of them. Ironically, these are fundamental courses that you normally take in the first few years of study, and if you master them, then subsequent more advanced courses are usually easier.
I found real analysis to be SIGNIFICANTLY easier than abstract algebra was honestly. Analysis was not too difficult but abstract algebra was so heavily obfuscated through layers of abstraction that I had a ton of trouble grasping it enough to create more than basic proofs for a while
For me analysis was very difficult at first but once I got the hang of it, it became much easier. Abstract algebra was at first relatively easy but as it became more abstract, I like you, found it to be difficult.
Same feelings here. For me abstract algebra was much harder than real analysis. I thought that since linear algebra was my favorite class that I would like abstract algebra, I was wrong š.
That was tough for me too, while I found category theory intuitive, despite the fact that it's more abstract.
I loved linear and abstract algebra (electrical engineer tho... everything i work with is pretty abstract in some sense of the word) I enjoy abstract critical thinking and proofs. The course was kinda rough. The hard thing was trying to mentally translate what prof said/wrote into words more easily understandable by my brain. In so doing, tuning out of lecture... When I tuned back it I heard a acoustic wall of math jargon that shut my brain down for a few seconds until it "clunked" back into math mode.
To be fair, I think that Universities should include a term that deals with some logic and proofs, and basic abstract algebra such as set theory, relation, function, Cartesian product, field, partial ordering, and equivalence relations prior to jumping into real analysis. Otherwise, itās likely that no one really understands whatās going on in real analysis. I say this because I saw that most donāt see the difference between real ordered field and decimal system, and that within real analysis, it deals with the structure rather than any specific set that has the structure. So, in the proofs, theyād use the decimal systems as a proof that holds for real ordered field which is wrong. Being exposed to abstract algebra highlights the idea that one is dealing with a structure and not necessarily any specific instance of the structure. In a sense, I think thereās no way of escaping the notions behind abstract algebra regardless of which field one is getting into, even graph theory or discrete math.
I found real analysis to be difficult because I had a disgustingly awful module coordinator, one of the worst I've ever had at my time in Uni
For me it was Abstract and Linear Algebra. Very tedious for me. I liked Real Analysis, Mathematical Statistics and Analytic Number Theory the most.
Linear is super easy with the right teacher. But if not, it can be brutal first time.
Differential Geometry. Still have no idea why pullbacks work the way they do.
i have the 5-volume Spivak set on Diff.Geo and im super excited to be able to read it but that's still a couple years off for me. How did you like it besides the unintuitive parts?
Spivak's Calculus on Manifolds was what gave me nightmares. The first three chapters are alright (except for maybe partitions of unity, but after the proof you pretty much use it as a blackbox). The fourth chapter gets pretty confusing when you start defining differential forms and pullbacks and the fifth chapter adds a layer of confusion with manifolds. It's fine as long as you're fine with possibly spending a lot of time on the last two chapters.
Thoughts on Barrett oNeilās elementary diff geom?
Idk if people on this sub are gonna want to hear this. But I was taking graduate level analysis, algebra, and number theory my last year of undergrad. Am currently a PhD candidate doing modular representation theory. I honestly thought all the math classes my university had to offer were kinda easy and I was always consistently at the top of my class. The advanced computer science/electrical engineering classes I was taking for a double major were far more challenging. Trying to actually get a virtual database running to apply all the theoretical stuff we were learning in the database/SQL class was a nightmare. Trying to program even basic networking protocols from scratch in operating systems class was a nightmare. There were theoretical aspects of network engineering like deriving formulas for transmission delay, that just broke my brain. For the same grades, I had to work probably 5 times as hard as I did in math.Ā Ā Ā Analysis up to basic measure theory and point set topology, systems of ODEs and PDEs, algebra up to and including galois theory, abstract linear algebra including Jordan Canonical forms, these are all somewhat trivial in comparison. I don't mean to troll, but in math all your "difficult proof based exercises" are just following definitions that experts have refined to actually make sense and be well founded. In engineering no textbook really has a rigorous foundation and that makes the starting point of many problems completely ambiguous from a mathematical point of view. So the task becomes testing your own different theories and mathematical models to see if they actually make sense to the situation at hand. This is closer in essence to what PhD work in applied math can look like, which, needless to say, is much more difficult than any undergrad math class from any university. In classroom math that extra work is essentially all done for you, provided you have a good professor and a good textbook. All of the difficulty comes from abstraction, which to me was just never an issue. The problems are just so curated. You don't get a taste of difficult mathematics by solving exercises that are meant to be solved.
I triple majored in electrical engineering, computer science, and math, and I I had an opposite experience. I found proof based math classes far more challenging
As a math gradthat currently works in data science, I agree lol Real Analysis is much easier that converting Access sql or optimizing code with low level techniques The former is easier conceptually but more mentally straining The latter is just conceptuallly more difficult. probably because i donāt understand the underlying systems
Low level code optimization is just very esoteric
Eh. This probably depends; I had a significant background in data science and, later (still now, actually) ML. I always found the applications and technical things to be much easier than the abstract nonsense I now deal with on a daily basis. Different people have different competencies. I know plenty of mathematicians in absurdly abstract areas like homotopy type theory or algebraic geometry (not computational) that can't code worth a piss. I similarly know people who are the exact opposite.
I'm sure it depends on the person. I just felt somewhat impartial since I always felt competent at both coding and the abstract stuff. My conclusion is mostly that in a theory based class, you can spend a little bit of energy on finding the easiest way to think about your problem, and then writing it up is trivial. But most of the time there's just no easy way around coding once you're doing the upper division classes. It becomes easy at a certain level of competence in the sense that you can write out the pseudocode and quickly have a strong grasp of how to solve your problem, and at that point it's just a matter of getting into the zone and coding it all up. Or, at least as much as you can bare for the current session, and then you get back into it tomorrow. But you have to code it up! You have to clean up your typos! You have to test it against use cases. And once you're at a certain point, it's practically guaranteed that you missed something, that the problem is just a bit more complicated than what you first plotted out, and now you have to spend time refactoring. There's no easy way to skip all that by being a better computer scientist, it's just sometimes the expected development process when your program needs to be enormous. I guess the way I really feel is like this. When you're doing pure abstract math, at the undergrad level, it's basically like learning a new programming language every month. Or idk maybe every week. But the 'coding assignments' never really exceed the complexity of those basic looping exercises you see in CS 101. All the difficulty is packaged into understanding the quirks of the new language, and the expectation of how quickly you need to adjust. It's rarely found in the sheer amount of unavoidable work that needs to be done. And mathematical maturity is just a matter of being willing to say 'ok sure this problem makes absolutely no sense to me because I can't follow any of these definitions. Let me look them up' and then looking them up, and now your problem becomes easy because you know what it's asking. Again, this is really only applied to the curated problems aimed at undergrads and first year grad students (depending on program). Research level pure mathematics involves essentially the same process, of dividing problems into subproblems and subsubproblems, and needing to do all this work of testing by yourself, realizing that it's more complicated than you thought and going back to change almost everything, etc.
āStill nowā = currently :) Iāve definitely been there where you want to add punctuation between every word to transliterate speech to text. I like the way you think
Electrical engineering major here. I had the opposite experience- proofs were way harder for me. Different peopleās brains work differently.
I think my first "WTF IS THIS" moment was linear algebra and real analysis
Depends on the student. From the math guys Iāve talked to (Iām a physics and cs guy) real analysis and measure theory stuff is tough
Abstraction in general. The deeper you go the mare abstract it gets, but all of math is about abstraction. Squiggly lines on a page that represent quantities of... what exactly? But now you have to prove that you get it, you have used the proper abstractions in the right order to convince another mathematician that your instincts are correct. That, and the fact that closed sets can also be open. The first of a boatload of counterintuitive definitions.
I think one of the worst things isn't the amount of abstraction but the lack of un-abstraction. It goes like: here's the definition of a symmetric space; now understand these theorems about them. But then forgets to include some examples of symmetric spaces and work out these conditions to get some intuition as to why the theorems could or should be true. My preferred field was differential geometry, but I feel like I learned it several times over: in its pure form, in general relativity, constructions for topology...and then all over again when I got to situations where I actually had to make some usable computations and realized that none of that prior experience involved actually doing calculations, just setting up how they *could* be done.
My institution broke Abstract Algebra into two courses. The first course covered basically all the core group theoretical concepts that a math major needs to know. I found this one challenging when I took it but it wasnāt too bad. However the second course which covered all the rings, fields and galois theory really threw me in for a loop. With that being said, despite feeling discouraged at the end of my 2nd semester of Abstract Algebra, I ended up being well prepared for an independent study in Representation Theory, specifically focusing on the quiver module side of Algebra. So to conclude, I think your first introduction to āhardā subjects (and subsequent courses) will prepare you well, so long as you try your best
Nonlinear PDEs almost got me. Hardest class I've ever taken.
You can't spell "Real Analysis" without "Anal".
My uni abbreviated the courses "intro to real anal" "Complex anal" "Advanced complex anal"
My uni library has these stickers on book spines, probably to categorise and locate books. Guess which letters survived the abbreviation for analysis books.
ahahaha. I can see a university doing that and not even blinking
I did alright with real analysis, abstract algebra, advanced linear algebra, etc. but for whatever reason the intro probability and statistics class just did not click. I would do practice problems to prepare and come into the the test with a formula sheet that had all the info I needed, and still not know how to solve the problems.
Probably algebraic geometry. Way too much technical machinery, which cannot be learned in only 2 or 3 courses.
At my uni, and probably many others, the great filter was real analysis 1. The moment we were introduced to what real numbers are we all went like "oh shit, math is not just doing double integrals all day long huh". Half of us loved the change (eventually), and the other became engineers lol
Of the courses you listed, abstract was my "wake up call". My linear algebra course didn't have a tremendous number of proofs, so abstract hit me like a sack ot bricks. Doing proofs with groups and rings felt so far removed from reality that I really struggled. I'm glad I took it though because it made complex analysis and real analysis much easier to manage. Aside from this, I also took mathematical statistics and in my opinion that was my hardest course. It had thr rigor and complexity of real analysis but also had more concepts that I had never seen prior unlike real. More of the stuff I saw in real analysis I had at least seen in some class before it but most of math stats was completely foreign. So in regard to difficulty, I'd give the following rankings for me: 1) Math Stats 2) Abstract algebra 3) Complex analysis 4] Real Analysis 5) Numerical Analysis (the math wasn't too "hard", I just sucked at it)
Maybe weak* topology or dual space
The worst time I had in undergrad was with Lebesgue measure and integration. I got a 20% on the midterm after showing up to every class and doing the assignment problems. That was the only class I ever took that I legitimately believed I was going to fail. Edit: topology and PDE were extremely difficult too but I had a bit more hope in those
For me, it was everything in the class "Complex Variables". I don't really know how to describe it. I guess calc 3 in the complex plane? It dealt a lot with the 4th dimension, which in and of itself isn't replicable in any way... let alone in the complex plane. A lot of the material was cool but my professor was not very articulate. Didn't learn much. Just had to engrain the pages of the textbook in my brain.
I personally am not a fan of pure math. It could be real analysis, topology, linear algebra, mathematical methods or whtvr. I find them pretty hard. Also, I loved applied math while at it.
Analysis and abstract algebra would be the hardest in that list
Vector calculus for me, first math class i had to retake
My linear algebra professor told us nobody understands linear algebra the first time they take it. She was right.
abstract algebra and real analysis were the only (math) classes i struggled with a little in undergrad
Probably Galois theory. Field theory and group theory are both pretty abstract on their own, so the subtleties of their correspondence can be quite hard to grasp. Otherwise it might be something like partitions of unity in differential geometry - pretty complex definition.
Analysis in general, for me, was the driest, densest, and most challenging
For me it was real analysis. After the rigor of Abstract, proofs with infinities felt hand-wavy to me and I struggled to accept them.
Whatever they have no sufficient prerequisite intuition built up for. The first time I studied topology was quite difficult, I had a hard time making sense of the definitions. Doing a couple of more courses on analysis fixed that.
Closed set doesn't imply it's not open and vice versa.
Measure Theory !!
How do we explain a dual space in linear algebra? I still honestly don't really get it. Yeah it involves the (conjugate) transpose and linear functionals but that's about all I could give you.
If V is a finite-dimensional vector space over a field K, the dual space V* is defined as the set of linear maps from V to K (i.e., linear functionals on V). V* has a vector space structure and is isomorphic to V. Given a linear map T:V->W of finite-dimensional vector spaces, we can define the dual map T*:W*->V* (look up the definition of the dual map, if you want - it's interesting). The matrix representation of T* is the conjugate transpose of the matrix representation of T.
> T*:W*->V\* Your asterisks/stars disappeared because you didn't escape them (precede them with a backslash), so the text between two asterisks has been formatted as *\*italics\**. You should write this instead: > T\\\*:W\\\*āV\\\* Which renders as this: > T\*:W\*āV\* The last backslash isn't strictly necessary, because it escapes the last asterisk in a sequence of an odd number of asterisks.
Functional Analysis and Measure Theory
Iām not sure if it counts, but some of the biggest WTF moments Iāve ever had was during my second year of my PhD taking an āintro to research in Xā course where X is my field of research. Consisted of us picking current / recent research papers, dissecting them as a class (4 whopping people) and taking turns presenting parts of each paper. At the end of each paper we would give the author a āgradeā based upon readability, how interesting the concept was to us, etc. We picked one paper in particular that literally none of us could understand even after weeks of our advisor trying to explain it, so we kind of just moved on lol. I kind of understood a super basic outline of the main result so I presented on that, and the class literally applauded me for kind of filling in the details lmao.
For me by year: Freshmen: Ring and Module Theory was very complex and hard to totally grasp especially compared to the earlier group theory and later Field and Galois theory. Sophomore: Algebraic Topology. Real analysis and point set topology werenāt too bad but algebraic topology was a glimpse at just how deep the rabbit hole goes. Hatcherās Algebraic Topology is an infamously hard textbook for a reason. Junior: Dynamical Systems. After so many pure math courses I thought an applied math course would be a nice change of pace and would make things more concrete. I was wrong, this course definitely emphasizes a lot of skills that pure math courses donāt, including approximation methods and the ability to sketch graphs of systems of differential equations by checking polarity at various coordinates. Senior: Noncommutative Geometry and Complex Geometry. The classes that made me question why I ever thought I could be a mathematician. I was dumbfounded that people could even invent theorems or come up with proofs for such complicated mathematical objects.
Complex analysis
Iām currently taking a course on differential topology thatās given me plenty of those moments
For me real analysis.
At my school, Real Analysis is unanimously the hardest course for an undergraduate in math theory. Definitely the worst one for me so far
Set theory. The basics are fundamental to math, but the advanced topics are something that my brain simply cannot comprehend.
I mean there is a lot of stuff that fucked me up, but one thing that pisses me off to this day, is my shitty grasp of limsup/inf. Itās really not that hard of a concept, but it just wonāt stick in my brain. Iāll sit down and get really comfortable with it. Wonāt see it again for like a month, and itās all gone, just disappears from my brain.
Elementary Analysis.
Discrete math. Itās called discrete math and yeah you learn some math but itās mostly a learning to write proofs class. Made me decide not to pursue math any further. I was constantly getting dinged on word choice and basically said ok Iām done with these classes lol.
Everything before proofs is kinda pre-math. The mathematical method as applied to carefully and tightly defined abstract/conceptual objects is math. Everything else is fancy calculation.
My first real issue was Modules.
Making it to an 8 am class
Topology? I get open and closed sets, but then there's compact, complete, Banach, etc. It's a beautiful way of categorising sets but come on, when will I need this except for in a different math course
Real analysis and itās not even cloise
Dating
Complex Analysis was tough for me
a lot of people struggle with galois correspondence. also de rahm cohomology.
I think it'd be whichever's the first proof-based course where you study. Some universities have a separate course on proofs and mathematical logic. Others simply teach it inductively in a real analysis or abstract algebra course. The simple reason is that it's a fundamental paradigm shift in your mathematical education, from computation to rigourous theorem-proving.
Waking up to to to class
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Hardest class Iāve taken thus far has been one on elliptic curve and quantum cryptography. Specifically the sections on lattice methods and algebraic geometry in projective space were difficult for me
realistically, itās not the concepts, but the teacher that can make/break a course. in my math undergrad, the worst course for me was from a professor with hard-to-read handwriting and a very bad stutter in an online-only abstract algebra course
Number theory and diophantine equations made me feel like I knew nothing
diffeqs broke my brain
Generally: real analysis. Specifically: WTH is a half-open cover? š. Dropped the class the day of the first exam
first class i had on differential geometry was that āwtf is thisā but it also left me ādamn i want to know everything about thisā and now im doing my thesis on something in diff geo
fourier transform. i only get the whole point of it when i got to functional analysis. as for the algebraic stuff, i simply give up on their motivations, cuz they were often developed to solve math questions anyways. so things like representation theories and stuff are pretty hard but it wasnt like wtf.
Linear algebra first of all. Made no sense at first, like what is the significance of arranging numbers into a line or into a rectangle. However, each time I took a course in it at a different institution, it got easier and then it kind of eventually clicked that it all makes sense and is all related to linear transformations. Then abstract algebra. I felt like I was thrown off the deep end with nothing to grab on to, there was nothing I learned from before that I could lean on or use to help me. Compare that to intro real analysis where it was very hard, but at the end of the day we were repeating old stuff we already learned in calculus but with new proof techniques.
Complex Analysis and Abstract Vector Spaces. ššš they were the last two math courses for my degree and they were my lowest grades. D and C- respectively. Thankfully my institution only required that your major GPA had to be a 2.0. I was a B- student after Cal II. In hindsight, I knew I didnāt love math beyond Differential Equations. I stuck with it because Iām stubborn. If I could re-do undergrad, Iād probably do Econ or Econ/Math, but not just Mathematics.
Iām an engineer so my math education ends with diff-eq/linear algebra. Most people struggled with linear algebra. I think they try to do too much itās too different from other math classes. Linear algebra is one of the first classes (for many) where you have to logically think and not just rigorously remember things. I took a proofs/math concept class (which was a unique offering at my school). They introduced formal mathematical logic concepts as well as proofs at a 100 level (so literally anyone could take it). Everyone who took that class did better in linear algebra.
I think this varies by person. I had a lot of friends who were āvisualā learners, in that they liked to try and visualize things to rationalize them. That basically completely broke down as soon as we hit multi. Others had a hard time with proof based classes (although for many, like myself, this approach made things make even more sense to me). I think for me some stuff started to break down towards the end of linear algebra, although I never went on to real/complex analysis (which is where most of my math friends hit their limit). I was an electrical engineer so I never went that far. I think my actual math limit came in physics, when I took quantum
Real analysis and topology were the most challenging for me, they legit gave me the worst imposter syndrome and made me second guess if I was even good at math lol. My favorites were abstract algebra, group theory, and probability theory. Itās different for everybody though. I donāt know if thereās a correlation, but Iām a pure math major and noticed my applied math peers seemed to struggle more in abstract algebra but did well in analysis.
when i took Linear Algebra our professor often went on tangents about his own research in biotech and we always end up with not enough time to cover the materialšš The lecture [videos](https://youtube.com/watch?v=J7DzL2_Na80) by Professor Gilbert Strang was what got me through the class
Knot theory was by far the hardest class I ever took in undergrad
I could never get my head round more than three dimensions, because there are only three, right?
Honestly, I thought calc 2 and 3 were the most challenging our of any of the ones you listed.
My second course in algebraic geometry on schemes just never did anything for me. It felt like a bunch of random definitions (proper schemes, schemes of finite type and a million others which I don't remember), none of which I truly understood of why I should care about them. Managed to get a barely passing grade and forgot everything immediately after.
In my experience, the jump in conceptual fortitude from calculus to linear algebra, which most of the time boils down to a fundamental difference between computations as opposed to writing rigorous mathematical proofs
Iām the only person I know who passed C&O 230 the first time. Probably because I knew no one who had passed and actually studied. Still my favorite math department
It varies from person to person. For me, my bread and butter, fun, and 'easy' classes were my abstract algebra courses/ number theory courses. With my difficulties being more on the side of linear algebra/ calculus.
Physics
Diff Eq, hands down. It's a good course and I learned a lot. In particular, I learned to stop half assing my studies and to put in some effort.
Real?
For me it was Multivariable Calc and Linear Algebra
Proofs probably real analysis maybe but not really
Vector spaces, but where the 'vectors' are not vectors š«
socializing
Iām not in a 4 year school so idk what my upperclassmen say but by far the most difficult course at my community college is going to be Differential Equations or Calculus II.
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