Linear Algebra before Calculus?

In summary: It's a bit of a chicken-and-egg problem. Linear algebra is a very abstract subject, and until you have a good understanding of that abstraction, it's difficult to see how linear algebra can be applied to real world problems. This is why linear algebra is usually taught second, after calculus has been covered a bit more concretely.
  • #1
zodiacbrave
11
0
Hello,

This might sound like a dumb question but can one learn Linear Algebra before Calculus?

Thank you.
 
Physics news on Phys.org
  • #2
Yes. My first course in linear algebra required no calculus.
 
  • #3
Definitely. I did a course on it without any background in calculus as well. What's interesting is that some results in linear algebra would also be used in a course on differential equations.
 
  • #4
My LA class had a few integrals and derivatives, but that as just for proofs, which we didnt need to know.
 
  • #5
The only time I ever used integrals and derivatives in my Linear Algebra course was to prove some things about Differential Equations. These were just examples though; they were only applications of Linear Algebra.

None of the proofs of Linear Algebra theory require any knowledge of Calculus, so except for some examples of applications, you will not need to know how to do derivatives or integrals at all (though some stuff that you might do later in the course may require some knowledge of the properties of the real and complex numbers)
 
  • #6
I'm quoting myself from the linear algebra forum:
Fredrik said:
You don't need calculus. You need to understand what functions are and have some idea what sets are to understand the definition of a vector space, and you need know how to add and multiply real numbers (e.g. rules like (a+b)(c+d)=ac+ad+bc+bd). But that's it.
 
  • #7
As already stated, linear algebra is completely independent of calculus. Algebra deals with discrete quantities, not continuous ones.

You can successfully learn linear algebra without any knowledge of calculus. The only problem may arise in applications of linear algebra, such as viewing the integral as a linear map or differential equations. In any case, these are tiny fractions of the whole subject.
 
  • #8
yes linear algebra is actually prerequisite for calculus done right. but as courses go in the us, we usually teach calculus as a collection of computational techniques, and then teach linear algebra more abstractly. so we teach linear algebra second because it si thought more difficult to understand "abstract ideas" than computational ones.

in an ideal world linear algebra is taught first, then calculus is taught as an application of linear algebra.

i.e. correctly done, calculus is the art of using linear algebra to deduce things about non linear functions.

e.g. the inverse function theorem in calculus says: if the derivative of a smooth function f at a is an invertible linear map, then locally near a, f is an invertible smooth map.
 
  • #9
mathwonk said:
e.g. the inverse function theorem in calculus says: if the derivative of a smooth function f at a is an invertible linear map, then locally near a, f is an invertible smooth map.

The problem with that is that "f is an invertible linear map" doesn't give me any geometric intuition. On the other hand, consider the more "calculus" explanation: that the tangent "plane" at a does not contain any of the axes, so in some area around near a, the function must be a bijection. I can actually visualize this, intuitively understand it, and think of how potentially do a proof.

Edit: Certainly, I could try picture every linear map as a geometric object, but I don't yet have that mathematical intuition, and it currently inhibits my ability to intuitively understand the abstract idea.
 
Last edited:
  • #10
mathwonk said:
i.e. correctly done, calculus is the art of using linear algebra to deduce things about non linear functions.

This.
 

1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the study of linear transformations, matrices, and systems of linear equations.

2. Do I need to know Linear Algebra before learning Calculus?

No, it is not necessary to have a thorough understanding of Linear Algebra before learning Calculus. However, some basic concepts such as matrices and vectors are often used in Calculus, so having a basic understanding of Linear Algebra can be helpful.

3. What are the main topics covered in Linear Algebra before Calculus?

The main topics covered in Linear Algebra before Calculus include vector operations, matrices and their properties, systems of linear equations, determinants, and eigenvalues and eigenvectors.

4. What are the applications of Linear Algebra in real life?

Linear Algebra has numerous applications in various fields such as engineering, physics, computer science, and economics. It is used to solve systems of linear equations, model and analyze data, and create computer graphics, among others.

5. Is Linear Algebra difficult to learn?

The difficulty of learning Linear Algebra can vary for different individuals. Some concepts may be challenging at first, but with practice and understanding, it can become easier. It is important to have a strong foundation in algebra before learning Linear Algebra.

Similar threads

  • Linear and Abstract Algebra
Replies
5
Views
1K
Replies
10
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
965
  • Linear and Abstract Algebra
Replies
4
Views
239
  • Linear and Abstract Algebra
Replies
2
Views
3K
Replies
20
Views
2K
  • Linear and Abstract Algebra
Replies
19
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
799
Back
Top