Is linear algebra harder than calculus? (important keys)

If you ask which is harder linear algebra or calculus, this post is for you.

In this article, we’re going to respond to these questions.

Also, discover the 10 hardest subjects that students study in linear algebra and the other that is in calculus. So keep reading to not miss-steps

Is linear algebra harder than calculus?

Calculus is harder than linear algebra, especially talking about calculus 3. Students study tough subjects like multivariable and complex theorems. At the same time, linear algebra requires dealing with many vectors and matrices, which is also hard. But calculus remains the harder one.

Calculus requires a lot of analysis and computation. To clarify, you need to make a lot of operations to functions such as derivatives and integrals. So it is noraml that responding to one calculus question could take you 1 or even 2 pages of writing.

Linear algebra is different. It tends to be cleaner by focusing a lot on geometry and vectors. In linear algebra, you will use intensively matrices that are not quite simple.

For this reason, we find some people saying that linear algebra is harder than calculus, whereas others say calculus is harder than linear algebra. So the question depends on where students find themselves or feel comfortable.

Calculus has its own hard subjects and linear algebra too. So this is what we’re going to discover in our next paragraphs. So keep reading to know what the hardest subjects you will study are in calculus and linear algebra.

These 5 subjects in linear algebra are harder than calculus.

1 – axioms

In axioms, you should learn how to make some operations in vectors, such as additions and multiplications.

In other words, how to multiply vectors such as scalar multiplication axioms. Also, how to multiply a vector with a scalar like u=3xV or something like that.

You will find a lot of students saying that linear algebra is harder than calculus just because they struggle in axioms vector spaces because they get confused about how to apply its rules.

2 – theorems for vector spaces

In linear algebra, there are many theorems for vector spaces that you should understand.

For example, one of the hardest things in linear algebra is the importance of showing them lots of vector spaces that aren’t just R^n. As a result, to see (for example):

  • the polynomials of degree at most n,
  • the continuous functions on some domains,
  • if matrices over R or C are vector spaces.

Also, many students struggle with the notion of linear independence and find hardships in solving geometrical representations.

Linear algebra requires thinking abstractly to grasp these concepts.

3 – Change of basis

Changing basis allows you to convert a matrix from a complicated form to a simple form. In other words, it is a method to simplify matrices.

For people who struggle with matrices, they say that linear algebra, especially on a change basis, is harder than calculus.

change basis is a subject that is also related to calculus because this technic can allow you to solve differential equations that are used in:

  • quantum mechanics
  • engineering such as electrical and mechanical engineering

4 – eigenvalues and eigenvectors

Many students find it difficult to determine eigenvalues and eigenvectors of a matrice. It gets hard to visualize and figure out the eigenvectors if we go above 3 dimensions.

Even in some cases, in 2d or 3d, it’s not that easy. So many people start to worry about these things in linear algebra.

Matrices are hard, and not very intuitive. we can make a lot of operations to matrices, such:

  • rotating
  • skewing
  • stretching
  • flip vectors

So it really difficult to find out what a matrix does just by looking at the numbers of the matrix.

This topic is used in mechanical engineering applications to study the solid deformations of objects. Evergreens are also used even by the Google ranking system.

You can read this article for more information.

5 – Jordan canonical form

Jordan canonical form is used to solve linear equations for computed and any square matric.

What makes Jordan’s canonical form harder than calculus is it is not feasible, which mean not being able to apply it in real-life application such as derivative or integral in calculus.

So it is hard to make any assumption between the reality and what you study in this theorem.

Theses 5 subjects in calculus are harder than linear algebra

Normally calculus is the hardest subject in math. Especially in calculus 3, things become serious and more complex than in calculus 1 or 2. You can read this article for more information about it.

Is calculus 2 hard than 3 ( solved and explained)

1 – multivariable

The first thing that makes calculus harder than linear algebra is multivariable. To clarify, multivariable calculus is a domain that studies functions with multiple variables like f(x,y,z,t) where you need to study and determine all aspects of these variables, such:

  • making a complex derivative means making a derivative function with more than 1 variable.
  • integrating multi variables functions

So you have to be good at derivatives and integrals to absorb well multivariable courses, meaning absorbing well the calculus precalculus and calculus 1 courses.

2 – Triple and double integrals

Integrals are a famous subject that makes many people suffer and say that calculus is hard than linear algebra, which is true. To calculate double and triple integrals, you must have a solid base in derivatives and original functions application.

Physicists use triple and double integral in many applications such as surface and calculation volume.

Things can become very messy with one integral operation it can take you two pages to solve one single operation. So calculus is not clean like linear algebra.

3 – vector field

In terms of the vector field calculations, they are not so hard and the ideas are easy to understand as well. But the proofs of these ideas or theorems are very hard.

if you don’t know what is vector field you would watch this introduction video.

The important thing that students find or struggle with is to visualize vectors in the long term behavior. So it is a bit tougher and requires hard work practicing a lot of examples.

4 – partial derivatives

Normally partial derivatives belong to the multivariable that we already talked about. Partial derivatives are a very confusing topic for many students.

What makes partial derivatives hard is you make a derivative of functions that have more than 1 variable. it could be 1, 2, or 3, or even more.

So making mistakes is highly probable, requiring being careful when you calculate partial derivatives.

5 – stoke’s theorem

to know the difficulty of the stoke theorem, I will let you with this video to watch below to see how the stokes theorem might be confusing for some people

Conclusion

Calculus and linear algebra have common subjects, especially multivariable. In other words, to learn or understand multivariable in calculus, you will need to know linear algebra to.

So calculus and linear algebra are not separated subjects. They have correlated relations with each other.

All the calculus topics we mentioned in this part belong to calculus 4, so if you want more info, you can read this article.

What is calculus 4? (explained in videos)

if you want to know more about the relationship that calculus has with algebra, you can read this article.

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