# Applied vs Proof Based Linear Algebra

• Courses
• bentleyghioda
In summary, the two courses described differ in their emphasis. The first course, Math 232, is more focused on applications, while Math 240 is more focused on proofs. However, both courses are essential for physics majors.
bentleyghioda
Hi, I’m going to be entering my first year of University this fall to study physics. In my second semester I will have to take a linear algebra course; however, my school has two different lower level linear algebra courses, and I must choose one. One course is focused more on applications of linear algebra, while the other is more focused on proofs. These are the descriptions of the two courses:

MATH 232: Linear equations, matrices, determinants. Introduction to vector spaces and linear transformations and bases. Complex numbers. Eigenvalues and eigenvectors; diagonalization. Inner products and orthogonality; least squares problems. An emphasis on applications involving matrix and vector calculations.

MATH 240: Linear equations, matrices, determinants. Real and abstract vector spaces, subspaces and linear transformations; basis and change of basis. Complex numbers. Eigenvalues and eigenvectors; diagonalization. Inner products and orthogonality; least squares problems. Applications. Subject is presented with an abstract emphasis and includes proofs of the basic theorems.

I am wondering if I would benefit more from the proof based class or the more applied class as a physics major. I do not have a lot of experience with proofs, but I will likely be taking a discrete mathematics course the same semester as linear algebra, and from what I’ve read, this course acts as an introduction to proofs. If it helps to know, I plan on going to grad school for physics, and I would also like to take several upper level math courses.

I would think the applied version would be more useful especially since you don’t have much proof experience. Proofs would be useful if you planned to focus on theoretical physics but even then applied uses would be more useful overall. Mathematicians benefit the most from proof based courses.

Klystron
The upper level math classes may be impossible if you do not have a strong background in proofs. If they are also applied math physics classes, that would be different.

jedishrfu
FactChecker said:
The upper-level math classes may be impossible if you do not have a strong background in proofs. If they are also applied math physics classes, that would be different.
So true, I blocked that traumatic memory until now. I once took an Abstract Topology course thinking I took proofs in geometry class so it should be no problem. It was definition on top of definition with no end in sight and I was a lost babe in a sea of upper-level math majors. Fortunately, my prof took pity on the hubris of a poor hapless physics major and guided me to a passing grade and implicitly warning to never tread into the ocean of no return without the proper math major background.

Thanks @FactChecker for bringing back that memory on my schooling of yesteryear.

gleem and FactChecker
Thank you for your replies. I’m not sure what courses I will be taking in the future, but I will keep in mind that many upper level math courses require proofs.

Do you think it would be beneficial to take some proof heavy math courses that are designed for math majors? For example, would I gain any tools from these classes that could be applied to physics?

If your ultimate goal is physics, I think you should learn your math from the physics perspective. Those classes should cover any of the theory that you need without getting hung up on every detail that formal proofs require.

Klystron and jedishrfu
I would take the intro proof based linear algebra... Although hard, you will walk out out from the class with a deeper understanding. Plus Linear Algebra is very important in Physics. You have 1 year to prepare...

@bentleyghioda the other question is whether you want to take a look at more formal mathematics. The safe, focused option is to go for the course more aligned to physics. Taking the proof-based course might be more adventurous and widen your knowledge of the maths-physics landscape.

Definitely take the application based class. It is a new subject for you and as a freshmen you should not overestimate your ability or over think your career. I think you will find it challenging for is my experience you will encounter much formal treatment of the subject while keeping you focused on why you are taking the course. It might be of value to find out which text each class uses and read the preface since it should contain the authors goals and methods

Klystron and PeroK

## 1. What is the difference between applied and proof-based linear algebra?

Applied linear algebra focuses on the practical applications of linear algebra in fields such as engineering, computer science, and physics. It involves using mathematical tools and techniques to solve real-world problems. Proof-based linear algebra, on the other hand, focuses on the theoretical foundations of linear algebra and involves proving theorems and properties.

## 2. Which one is more useful in the real world?

Both applied and proof-based linear algebra have their own uses in the real world. Applied linear algebra is more useful in solving real-world problems and developing practical solutions, while proof-based linear algebra is more useful for understanding the underlying principles and concepts of linear algebra.

## 3. Can one be proficient in one without the other?

Yes, it is possible to be proficient in one without the other. Some people may focus on one aspect of linear algebra, while others may have a strong understanding of both applied and proof-based concepts. It ultimately depends on one's interests and goals.

## 4. What are the key differences in the approach to learning and using applied vs proof-based linear algebra?

The approach to learning and using applied linear algebra is more hands-on and practical, often involving the use of software and technology. Proof-based linear algebra, on the other hand, involves a more theoretical approach, with a focus on logical reasoning and proof writing.

## 5. Which one is more important for a career in science or engineering?

Both applied and proof-based linear algebra are important for a career in science or engineering. Applied linear algebra is essential for solving real-world problems and developing practical solutions, while proof-based linear algebra provides a deeper understanding of the underlying principles and concepts, which can be valuable for more advanced research and development in these fields.

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