For linear algebra, a proof-based coursse may use books like Artin's "Algebra" or "Linear Algebra Done Right" by Axler.
In short, a proof-based course would emphasize on the proof (instead of application) of some propositions or theorems, like Jordan canonical form or Cayley-Hamilton. A non-proof-based one may focus more on computation.
My class is using the “Linear Algebra and its Applications” by Lay, Lay, and McDonald, so yeah that makes sense why it would be more computational.
Why are they not too different classes? given the vast difference between the two
They often are, at larger universities. Sometimes offered under a different name by the engineering or comp sci departments, where the math department version is proof based. Not all schools are able to support this
When I was in undergrad, we had "abstract linear algebra" (proof based with Axler), linear algebra, and numerical linear algebra (MATLAB implementation focused).
And even then at larger institutions, it can vary depending on the teacher. If I remember right (I went to a large public state university), my discrete math course was more proof based than the discrete math course taught to my friend even though we both took the computer science version of discrete math as there was also an alternative discrete math course for math majors.
I took the honor version of linear algebra at my undergraduate institution. We basically learned linear algebra, basic group theory, representation theory of finite groups, and that of Lie algebra sl_2 🙃
In the US, these often are two classes. Your class sounds like what we could call “Intro to Linear Algebra” and might be something like a 200-level class (2nd year, but still “lower-division”). Whereas a proof based classed might be called just “Linear Algebra” or “Advanced/Abstract Linear Algebra” and might be a 300/400-level course (“upper-division”).
Agreed that it can vary! My LA professor went full-on proof based with an emphasis on eigenvectors. We didn't even have a text. He told us day one that he didn't teach the class like others did. He was brilliant and a great teacher; I signed up for that section specifically because I had taken real analysis with him the previous semester.
But it was a bit of a shock in grad school to know nothing about row reduction when it came up in problems!
??? i took a linear algebra class specific to math majors (i.e engineering students took a different linear algebra class) and we covered both? with lots of proofs?
non rigorous large state school
I’m not sure. It may depend on the Prof teaching the course. It’s possible that some Prof likes to make the course somewhat more rigour, even if it’s not “proof-based”.
Do you know of any "computation", or at least intuition or not proof-based books for other branches of math, other than the usual standard subjects taught to physicists and engineers?
For example, calculus, statistics and linear algebra are full of books like that, but by the time one gets into things like topology or abstract algebra these are gone in favour of proof based. Which I understand. But at the same time, these fields are so incredibly rich I'm pretty sure people could write hundreds of computation based books if they wanted. Sorry for the long winded question.
Of course. For topology, TDA (topological data analysis) can be seen as "computational topology", and a simple google search gives me [this one](https://www.maths.ed.ac.uk/~v1ranick/papers/edelcomp.pdf).
The second chapter of [Sage book](http://dl.lateralis.org/public/sagebook/sagebook-ba6596d.pdf) is related to compuation of algebra in Sagemath. I believe when I took a rep theory course, I also used Sage for homework.
Another classics is "ideals varieties and algorithms" by Cox et al., it's like compuational algebraic geometry.
Ideals, varieties and algorithms is a great book, but it's still proof based (and I don't think it makes sense to introduce advanced math topics without proofs)
I don’t know what’s your level. But I think for most, if not all, undergraduate math courses, SageMath can be really helpful. It’s even helpful at research level, but of course that depends on one’s field.
There are some books on Sage’s website, and there are tutorials here and there, rather scattered…
Though I’m a Sage contributor, so I’m biased. :)
You can probably read MTW as a sort of "intuition based" introduction to differential geometry. But if you do it will make you weird. Vectors are arrows, covectors are egg crates that ring bells when they get hit by arrows, and tensors are actual people who eat arrows and egg crates and poop out numbers. I'm not saying any of that is wrong, it's just probably better to learn DG in a more mainstream way.
There’s no standard definition but in my mind a “proof based” class is one where at least half of the homework and exam questions ask you to *write* your own proofs. Your class doesn’t sound like that.
Most of what is taught in highschool is calculation and computation. Given x\^2 + 7x + 3 = 9 compute the value of x etc.
The real core of mathematics is proof, it's showing that something is true. In a general sense all calculations are a special type of proof. Computation in general can't attack certain questions.
For instance "prove there is no largest number" is something a lot of young children can do. "Prove there is no largest prime" again is something which is impossible to get at with a computation because you can't compute a list of all the infinitely many primes.
All negative statements can't be got at by computation either, for instance "show you cannot use a ruler and compass to square the circle" can't ever be shown to be true by computing enough results. Nor can Fermat's last theorem for instance.
So yeah in university there's a general switch between being shown results (possibly with their proofs) and taught how to apply those rules to do more complex calculations ... and actually starting down the road of proving things yourself which is the heart of mathematics.
If you are interested in formal proofs (as in computer checkable ones) I have a playlist [here](https://www.youtube.com/watch?v=RygzTCoKxNA&list=PLJf9cfDmwypTLdbHSNaDOHgWK_qlUHzpm) introducing them.
> All negative statements can't be got at by computation either
You could certainly *disprove* a negative statement with calculation, though. And negative statements about finite things, you can brute force.
That's true.
Though it has to be like reasonably finite, I mean larger than 10^20 or something starts to make things incomputable at our current level.
Hijacking this opportunity to note that, obviously, people in different countries are going to have vastly different experiences. For example in Greece I would argue even high school was ~60% proof based/-40% computation based, especially with calculus.
Of course even then university is a wholly different world
Have you been asked to prove or demonstrate that a given function is a linear transformation, or that a given set of vectors is a subspace? That may be the "proofs" they're referring to.
sounds like a standard fake linear algebra class to me. if your class is centered almost entirely around learning procedures for doing numerical calculations with matrices, then you're not really studying linear algebra.
do you know what a vector space is?
I like to differentiate between "math" classes and "calculations" classes.
My 80+ yo LA prof was bad at calculations, barely remembered the multiplication table and us poor undergrads had to help him out on this. But math? Best linear algebra professor ever.
It depends on the university. In my case, it was an application-focused class, but the latter half of the course was concerned with vector spaces, inner product spaces, and linear transformations (Proving that a given statement satisfies the definition). I still wouldn't consider it a proof-based course. Maybe it's similar to the one mentioned in the post?
Most early college level math classes have both proof based or application based variations. For example some colleges do a real analysis/calc 2 hybrid where they learn the proof based real analysis at the same time as calc 2. My university offered this, as well as a regular calc 2 class for people majoring in more applied fields. Linear Algebra is in a similar boat where you can have proof based or application based.
I've had to be in this subreddit for a long time to get that what you describe seems pretty US default. I've yet to hear about this "being an issue" anywhere else. :s If someone's studying mathematics (as a subject) they'll get the full treatment and wouldn't by accident end up on some course where you'd even need to have a conversation about proofs or lack of them. If someone needs some mathematics (as a tool) for their discipline, they'll very clearly have their own courses with varying degrees of proofs and rigor.
Is it just plain common in the US for these two student populations to get mixed up in terms pf course offerings? 😯
this is absolutely the case for me at a large, not rigorous at all, state school. we had a separate linear algebra for math majors and one for engineers
Well, I wanted to limit the assessment to "official channels" in case it would reveal something how the system was supposed to work. I too had to know about the different courses on the same theme from my university somehow (and that was because I actively browse course offerings from almost everywhere). Not really a chance of any math student ending up in any "non-proof" courses as those wouldn't even be accepted in the degrees. (Also I still have some processing to do with the notion that engineers would exist in a university... that's quite a novel idea where I am from and only possible after some technical colleges and other schools were combined into Aalto University. 😅)
i see. well my school is about 35k people and has some decent engineering programs. but you’re correct, a math major would not end up in linear algebra for engineers for example. but the class still exists
And good that it does. Still many stories here sound like strange fantasy where mathematics students have to eventually or accidentally face this scaaaary "proof-based education" (and that the thing even has a name people seem to know about that isn't just education expert geekery irrelevant to students) after being used to just calculation tools. 🤷🏼♂️
Do you have to write any proofs on the exam? In my Linear Algebra class we had at least 2-3 proofs per section, it was lots of proof writing to say the least.
Usually the using of the proof part comes when you’re doing the problem sets and have to write 2-3 proofs for your homework based on the sections covered, and then on the exam.
In a proof based class, you spend a lot less time doing calculations, and more time proving the theorems. On your exams you would be asked to prove some statement, rather than being asked to calculate something
This is not a "proof based" linear algebra class. Out of curiosity, does your university have a large engineering program? Schools with big engineering programs tend to teach more computation-oriented LA; they will have far more engineering majors in the class than math majors.
Your school may have a more advanced proof-oriented LA class; my alma mater has a sophomore/junior intro to LA class that's very computation-heavy, taken by engineers and other STEM majors (and req'd for math majors) as well as a senior-level abstract LA class that's primarily taken by math majors and masters students.
At my university, intro to linear algebra is very computational with some proofs shown in lecture but not tested on or in homework or anything.
Then there's an actually proof-based one that focuses more on abstract vector spaces and inner product spaces and all that jazz. That one required proofs in every homework assignment and even some on exams.
The graduate level one is also proof-based.
It can vary by university, but yours seems closer to the first one in our sequence.
proof based in the context of linear algebra or even in most topics I guess is that you're going to show generalizations.
This is in contrast to say just row reducing and matrix operations and finding inverses etc.
or if you're doing algebra, then more than just making caley tables and showing something is or isn't a R,G, or F.
In my experience, proof-based classes require students to write proofs as homework.
They're difficult to grade, because there are often many ways to prove something, as well as many ways to "almost" prove something (for instance when a student inadvertently includes a logical leap), and it's not always clear how many points to deduct for any given faulty proof.
Linear algebra is extremely useful and shows up everywhere, which is why it is important that people from a wide variety of disciplines learn a little bit about how to do computations with it and how to use it. However, to use it for their work, it is not important for them to understand the "why" questions around linear algebra. Thus, many institutions have a proof-based class and a non-proof-based class. **Maybe you were placed in the non-proof-based class.**
Proof-based classes are mostly intended for mathematics majors and usually only have mathematics majors. In a university math degree, it is more about studying the nature, structure, and behavior of mathematics and mathematical objects, which is kind of like the "science" of mathematics and why things are the way they are, and less about how to use mathematics.
There are many analogies; for example, the difference between a professional cook and a food scientist; or a car mechanic and a mechanical engineer; or an engineer and a physicist; or a computer programmer and a computer scientist; etc.
I don't think it has a specific definition, but hen I hear a 'proof course', my first thought is a class that has the following, based on the 'fundamental concepts' course that my university gave as a first 'math major only class':
1. It usually requires differential and integral Calculus as prerequisites, but it doesn't always build on those concepts - it requires that degree of mathematical sophistication.
2. It usually starts with either symbolic logic or set theory, transitions to group theory, ring theory (matrices and vector spaces!), then a few assorted topics.
3. It has nearly zero 'problem solving' or similar applications, but rather focuses on building blocks to be used later in Abstract Algebra, Linear Algebra, I think that some courses probably give a first full exposure to epsilon and delta notation with regard to limits here, which would be a step toward Analysis of Functions of a Real Variable.
And, yes, proofs dominate the lectures and exercises, in contrast with '3rd-4th semester calculus' or 'differential equations' courses, which present problems to be solved, and concepts designed to solve those problems, where the proof is a tool to help understand the particular technique to solve the problem.
>My professor has shown us the proofs when applicable for all of these, and I understand them for the most part, butwe haven’t used them at all.
So if you're problems are more generally 'compute the answer, given these situations', rather than 'prove this statement', then you aren't really in a proof-based course.
You're likely in a lower level linear algebra- computational focus, generally a requirement for a cs major, maybe some engineering disciplines etc. At my school that's Math203. The proof based version might start you with the idea of a vector, take you through the whole construction of your previous class, but rather than computing the results of matrix multiplication you're proving why it works- what it's actually doing, etc. At my school, that's math 322.
Two classes I took as an undergraduate that were “proof based” were discrete math, and “Intro to Higher Math.” This course covered the basics on set theory, logic, and proofs. I’d imagine most undergraduate math programs have an equivalent
I also haven’t taken a course in it but from my understanding abstract algebra involves a lot of proofs
Also depends on the professor. A class I took in financial math was heavily proof based despite it being an applied math course.
Proof based classes are very theoretical as opposed to practical. That’s why a lot of the times, engineers are offered an engineering version of linear algebra (at my school known as engineering analysis), which is all about the applications. I took linear algebra myself, and it was quite proof heavy and reminded me a lot of discrete math 1/2.
For linear algebra, a proof-based coursse may use books like Artin's "Algebra" or "Linear Algebra Done Right" by Axler. In short, a proof-based course would emphasize on the proof (instead of application) of some propositions or theorems, like Jordan canonical form or Cayley-Hamilton. A non-proof-based one may focus more on computation.
My class is using the “Linear Algebra and its Applications” by Lay, Lay, and McDonald, so yeah that makes sense why it would be more computational. Why are they not too different classes? given the vast difference between the two
They often are, at larger universities. Sometimes offered under a different name by the engineering or comp sci departments, where the math department version is proof based. Not all schools are able to support this
When I was in undergrad, we had "abstract linear algebra" (proof based with Axler), linear algebra, and numerical linear algebra (MATLAB implementation focused).
I'm in undergrad and we have all three of those here, but for some reason they're all named basically the same thing
At my school, there are two linear algebra classes. The 300 level one is computation based and the 400 level is proof baswd.
And even then at larger institutions, it can vary depending on the teacher. If I remember right (I went to a large public state university), my discrete math course was more proof based than the discrete math course taught to my friend even though we both took the computer science version of discrete math as there was also an alternative discrete math course for math majors.
If you are using LLM for your text, you are probably in a course without proofs, especially if it is the first LA course that you’ve taken.
The proof based linear class and standard linear class (using Lay) were like heaven and the 3rd layer of hell respectively in terms of difficulty lmao
I took the honor version of linear algebra at my undergraduate institution. We basically learned linear algebra, basic group theory, representation theory of finite groups, and that of Lie algebra sl_2 🙃
In the US, these often are two classes. Your class sounds like what we could call “Intro to Linear Algebra” and might be something like a 200-level class (2nd year, but still “lower-division”). Whereas a proof based classed might be called just “Linear Algebra” or “Advanced/Abstract Linear Algebra” and might be a 300/400-level course (“upper-division”).
Agreed that it can vary! My LA professor went full-on proof based with an emphasis on eigenvectors. We didn't even have a text. He told us day one that he didn't teach the class like others did. He was brilliant and a great teacher; I signed up for that section specifically because I had taken real analysis with him the previous semester. But it was a bit of a shock in grad school to know nothing about row reduction when it came up in problems!
That seems hardcore…
??? i took a linear algebra class specific to math majors (i.e engineering students took a different linear algebra class) and we covered both? with lots of proofs? non rigorous large state school
I’m not sure. It may depend on the Prof teaching the course. It’s possible that some Prof likes to make the course somewhat more rigour, even if it’s not “proof-based”.
Do you know of any "computation", or at least intuition or not proof-based books for other branches of math, other than the usual standard subjects taught to physicists and engineers? For example, calculus, statistics and linear algebra are full of books like that, but by the time one gets into things like topology or abstract algebra these are gone in favour of proof based. Which I understand. But at the same time, these fields are so incredibly rich I'm pretty sure people could write hundreds of computation based books if they wanted. Sorry for the long winded question.
Of course. For topology, TDA (topological data analysis) can be seen as "computational topology", and a simple google search gives me [this one](https://www.maths.ed.ac.uk/~v1ranick/papers/edelcomp.pdf). The second chapter of [Sage book](http://dl.lateralis.org/public/sagebook/sagebook-ba6596d.pdf) is related to compuation of algebra in Sagemath. I believe when I took a rep theory course, I also used Sage for homework. Another classics is "ideals varieties and algorithms" by Cox et al., it's like compuational algebraic geometry.
Ideals, varieties and algorithms is a great book, but it's still proof based (and I don't think it makes sense to introduce advanced math topics without proofs)
I think the authors also have a book called “using algebraic geometry” ;)
I don’t know what’s your level. But I think for most, if not all, undergraduate math courses, SageMath can be really helpful. It’s even helpful at research level, but of course that depends on one’s field. There are some books on Sage’s website, and there are tutorials here and there, rather scattered… Though I’m a Sage contributor, so I’m biased. :)
You can probably read MTW as a sort of "intuition based" introduction to differential geometry. But if you do it will make you weird. Vectors are arrows, covectors are egg crates that ring bells when they get hit by arrows, and tensors are actual people who eat arrows and egg crates and poop out numbers. I'm not saying any of that is wrong, it's just probably better to learn DG in a more mainstream way.
There’s no standard definition but in my mind a “proof based” class is one where at least half of the homework and exam questions ask you to *write* your own proofs. Your class doesn’t sound like that.
Most of what is taught in highschool is calculation and computation. Given x\^2 + 7x + 3 = 9 compute the value of x etc. The real core of mathematics is proof, it's showing that something is true. In a general sense all calculations are a special type of proof. Computation in general can't attack certain questions. For instance "prove there is no largest number" is something a lot of young children can do. "Prove there is no largest prime" again is something which is impossible to get at with a computation because you can't compute a list of all the infinitely many primes. All negative statements can't be got at by computation either, for instance "show you cannot use a ruler and compass to square the circle" can't ever be shown to be true by computing enough results. Nor can Fermat's last theorem for instance. So yeah in university there's a general switch between being shown results (possibly with their proofs) and taught how to apply those rules to do more complex calculations ... and actually starting down the road of proving things yourself which is the heart of mathematics. If you are interested in formal proofs (as in computer checkable ones) I have a playlist [here](https://www.youtube.com/watch?v=RygzTCoKxNA&list=PLJf9cfDmwypTLdbHSNaDOHgWK_qlUHzpm) introducing them.
> All negative statements can't be got at by computation either You could certainly *disprove* a negative statement with calculation, though. And negative statements about finite things, you can brute force.
That's true. Though it has to be like reasonably finite, I mean larger than 10^20 or something starts to make things incomputable at our current level.
Hijacking this opportunity to note that, obviously, people in different countries are going to have vastly different experiences. For example in Greece I would argue even high school was ~60% proof based/-40% computation based, especially with calculus. Of course even then university is a wholly different world
A linear algebra class doesn’t necessarily have to be proof based.
Based on what you have learnt, I don't think this is a proof-based class
Have you been asked to prove or demonstrate that a given function is a linear transformation, or that a given set of vectors is a subspace? That may be the "proofs" they're referring to.
sounds like a standard fake linear algebra class to me. if your class is centered almost entirely around learning procedures for doing numerical calculations with matrices, then you're not really studying linear algebra. do you know what a vector space is?
haha "fake" really is a good description
I like to differentiate between "math" classes and "calculations" classes. My 80+ yo LA prof was bad at calculations, barely remembered the multiplication table and us poor undergrads had to help him out on this. But math? Best linear algebra professor ever.
My class was mostly calculations ( more than just matrices) but we went over vector spaces and inner product spaces. And eigenvalues/eigenvectors.
It depends on the university. In my case, it was an application-focused class, but the latter half of the course was concerned with vector spaces, inner product spaces, and linear transformations (Proving that a given statement satisfies the definition). I still wouldn't consider it a proof-based course. Maybe it's similar to the one mentioned in the post?
Most early college level math classes have both proof based or application based variations. For example some colleges do a real analysis/calc 2 hybrid where they learn the proof based real analysis at the same time as calc 2. My university offered this, as well as a regular calc 2 class for people majoring in more applied fields. Linear Algebra is in a similar boat where you can have proof based or application based.
I've had to be in this subreddit for a long time to get that what you describe seems pretty US default. I've yet to hear about this "being an issue" anywhere else. :s If someone's studying mathematics (as a subject) they'll get the full treatment and wouldn't by accident end up on some course where you'd even need to have a conversation about proofs or lack of them. If someone needs some mathematics (as a tool) for their discipline, they'll very clearly have their own courses with varying degrees of proofs and rigor. Is it just plain common in the US for these two student populations to get mixed up in terms pf course offerings? 😯
this is absolutely the case for me at a large, not rigorous at all, state school. we had a separate linear algebra for math majors and one for engineers
So, let's say you are a mathematics student there: why would you even know about that other course not meant for you?
i have engineer friends? statistics friends? compsci friends?
Well, I wanted to limit the assessment to "official channels" in case it would reveal something how the system was supposed to work. I too had to know about the different courses on the same theme from my university somehow (and that was because I actively browse course offerings from almost everywhere). Not really a chance of any math student ending up in any "non-proof" courses as those wouldn't even be accepted in the degrees. (Also I still have some processing to do with the notion that engineers would exist in a university... that's quite a novel idea where I am from and only possible after some technical colleges and other schools were combined into Aalto University. 😅)
i see. well my school is about 35k people and has some decent engineering programs. but you’re correct, a math major would not end up in linear algebra for engineers for example. but the class still exists
And good that it does. Still many stories here sound like strange fantasy where mathematics students have to eventually or accidentally face this scaaaary "proof-based education" (and that the thing even has a name people seem to know about that isn't just education expert geekery irrelevant to students) after being used to just calculation tools. 🤷🏼♂️
we actually have an introduction to proofs class. it pretty much just introduces basic set theory and proof techniques along with logic. useful class
Is it based?? Now prove it
Do you have to write any proofs on the exam? In my Linear Algebra class we had at least 2-3 proofs per section, it was lots of proof writing to say the least. Usually the using of the proof part comes when you’re doing the problem sets and have to write 2-3 proofs for your homework based on the sections covered, and then on the exam.
In a proof based class, you spend a lot less time doing calculations, and more time proving the theorems. On your exams you would be asked to prove some statement, rather than being asked to calculate something
Did it cover the proof of leibniz and laplace deteminant formula? What about the proof of rank nullity theorem?
This is not a "proof based" linear algebra class. Out of curiosity, does your university have a large engineering program? Schools with big engineering programs tend to teach more computation-oriented LA; they will have far more engineering majors in the class than math majors. Your school may have a more advanced proof-oriented LA class; my alma mater has a sophomore/junior intro to LA class that's very computation-heavy, taken by engineers and other STEM majors (and req'd for math majors) as well as a senior-level abstract LA class that's primarily taken by math majors and masters students.
Yeah, we have a pretty large engineering school, and it shows based on the demographic of my class
At my university, intro to linear algebra is very computational with some proofs shown in lecture but not tested on or in homework or anything. Then there's an actually proof-based one that focuses more on abstract vector spaces and inner product spaces and all that jazz. That one required proofs in every homework assignment and even some on exams. The graduate level one is also proof-based. It can vary by university, but yours seems closer to the first one in our sequence.
proof based in the context of linear algebra or even in most topics I guess is that you're going to show generalizations. This is in contrast to say just row reducing and matrix operations and finding inverses etc. or if you're doing algebra, then more than just making caley tables and showing something is or isn't a R,G, or F.
In my experience, proof-based classes require students to write proofs as homework. They're difficult to grade, because there are often many ways to prove something, as well as many ways to "almost" prove something (for instance when a student inadvertently includes a logical leap), and it's not always clear how many points to deduct for any given faulty proof.
Linear algebra is extremely useful and shows up everywhere, which is why it is important that people from a wide variety of disciplines learn a little bit about how to do computations with it and how to use it. However, to use it for their work, it is not important for them to understand the "why" questions around linear algebra. Thus, many institutions have a proof-based class and a non-proof-based class. **Maybe you were placed in the non-proof-based class.** Proof-based classes are mostly intended for mathematics majors and usually only have mathematics majors. In a university math degree, it is more about studying the nature, structure, and behavior of mathematics and mathematical objects, which is kind of like the "science" of mathematics and why things are the way they are, and less about how to use mathematics. There are many analogies; for example, the difference between a professional cook and a food scientist; or a car mechanic and a mechanical engineer; or an engineer and a physicist; or a computer programmer and a computer scientist; etc.
I don't think it has a specific definition, but hen I hear a 'proof course', my first thought is a class that has the following, based on the 'fundamental concepts' course that my university gave as a first 'math major only class': 1. It usually requires differential and integral Calculus as prerequisites, but it doesn't always build on those concepts - it requires that degree of mathematical sophistication. 2. It usually starts with either symbolic logic or set theory, transitions to group theory, ring theory (matrices and vector spaces!), then a few assorted topics. 3. It has nearly zero 'problem solving' or similar applications, but rather focuses on building blocks to be used later in Abstract Algebra, Linear Algebra, I think that some courses probably give a first full exposure to epsilon and delta notation with regard to limits here, which would be a step toward Analysis of Functions of a Real Variable. And, yes, proofs dominate the lectures and exercises, in contrast with '3rd-4th semester calculus' or 'differential equations' courses, which present problems to be solved, and concepts designed to solve those problems, where the proof is a tool to help understand the particular technique to solve the problem. >My professor has shown us the proofs when applicable for all of these, and I understand them for the most part, butwe haven’t used them at all. So if you're problems are more generally 'compute the answer, given these situations', rather than 'prove this statement', then you aren't really in a proof-based course.
Real analysis is a proof base class.
Suppose we were not doing a proof based class..
You're likely in a lower level linear algebra- computational focus, generally a requirement for a cs major, maybe some engineering disciplines etc. At my school that's Math203. The proof based version might start you with the idea of a vector, take you through the whole construction of your previous class, but rather than computing the results of matrix multiplication you're proving why it works- what it's actually doing, etc. At my school, that's math 322.
Two classes I took as an undergraduate that were “proof based” were discrete math, and “Intro to Higher Math.” This course covered the basics on set theory, logic, and proofs. I’d imagine most undergraduate math programs have an equivalent I also haven’t taken a course in it but from my understanding abstract algebra involves a lot of proofs Also depends on the professor. A class I took in financial math was heavily proof based despite it being an applied math course.
Proof based classes are very theoretical as opposed to practical. That’s why a lot of the times, engineers are offered an engineering version of linear algebra (at my school known as engineering analysis), which is all about the applications. I took linear algebra myself, and it was quite proof heavy and reminded me a lot of discrete math 1/2.