Math Books: Linear, Multilinear, Category & Graph Theory

In summary, the conversation discusses the need for resources in linear algebra, multilinear algebra, category theory, and graph theory, specifically at the graduate level. Several recommended books are mentioned for each subject, with a focus on rigorous and comprehensive texts. The idea of starting with undergraduate books before moving on to more abstract concepts is also mentioned.
  • #1
Tenshou
153
1
I don't know if this the way I should be going about this, but I need either websites or books. Mainly in Linear Algebra, multilinear Algebra, category theory, and graph theory. I mean It would be cool to find these types of books at the graduate level. I mean, I want to be a mathematician and the only book I have currently is a book on linear algebra. Also, why are math books so expensive, I found this book on multilinear algebra and it cost 175 bucks and then I found a book on category theory which cost about 120, talk about over priced... But I guess the material in those book must be top shelf material.
 
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  • #2
Not sure if starting with grad books is wise, but anyway...

For linear algebra, the most awesome book is Roman: https://www.amazon.com/dp/1441924981/?tag=pfamazon01-20 you can't get better than this. This also covers multilinear algebra.
Another good reference book is Greub: https://www.amazon.com/dp/3540901108/?tag=pfamazon01-20

For category theory, there is of course MacLane: https://www.amazon.com/dp/1441931236/?tag=pfamazon01-20
A very good (and legally free) book is the joy of cats: katmat.math.uni-bremen.de/acc/acc.pdf
A rather advanced, but extremely comprehensive work is the tome of Borceux: https://www.amazon.com/dp/0521061199/?tag=pfamazon01-20

I have no idea about graph theory.
 
  • #3
  • #4
Thanks micromass, and Ah, yes Werner Greub I have his Linear ALgebra Text already. I have actually been getting help on solving some of the problems in his book from this site :) Thank you all for that. Oh noes, undergraduate text... Thanks anyway, but undergraduate material is decent but not enough rigor for me :) D: someone has to know something about graph theory, I mean I am getting a book by bella bollobas (Such that I have a copy of my own.) but currently I do not have one in my possession, that book cost me a arm and a leg on the budget I am on(like most of those books lol) But thanks for the free source, even if it is at an undergrad level
 
  • #5
You seem to think that undergrad books are not rigorous. I don't know how you got that idea. Undergrad books are usually as rigorous as grad books. The difference between undergrad and grad is mostly that grad books are more abstract. For example, functional analysis covers Banach spaces and Hilbert spaces. But a grad book might do it in the setting of locally convex spaces. Both are as rigorous, but the level of abstraction is much higher in grad books. Jumping in the highest possible abstraction immediately is not recommendable.

If you read baby Rudin, or Axler, then both are undergrad books but they are both very rigorous. Most books are like this.

Although I can read most grad books, I always like to look at undergrad books. The concepts are much clearer there, and the motivation is given. Grad books sometimes look like just an unmotivated list of results.
 
  • #6
This maybe true, but the abstraction is interesting to try to conceptualize. Like to conceptualize a finite graded linear space of two duals and a homogenous homeomorphism between them of degree zero(A problem I am currently stuck on lol) I guess that is some what abstract. But still, the level of thought is different from undergrad text.
 
  • #7
Tenshou said:
This maybe true, but the abstraction is interesting to try to conceptualize. Like to conceptualize a finite graded linear space of two duals and a homogenous homeomorphism between them of degree zero(A problem I am currently stuck on lol) I guess that is some what abstract. But still, the level of thought is different from undergrad text.
I think this post indicates why one should not jump to graduate level.

Stick with UG books until you are ready to abstract.
 

Related to Math Books: Linear, Multilinear, Category & Graph Theory

1. What is the difference between linear and multilinear algebra?

Linear algebra deals with vector spaces and linear transformations, while multilinear algebra extends this concept to deal with tensors, which are objects that can have multiple dimensions and can represent more complex relationships between vectors.

2. How is category theory related to mathematics?

Category theory is a branch of mathematics that studies the structure of mathematical systems and the relationships between them. It provides a framework for understanding and organizing various mathematical concepts and structures.

3. What is graph theory and how is it used?

Graph theory is the study of graphs, which are mathematical structures that represent relationships between objects. It is used in many fields, such as computer science, biology, and social sciences, to model and analyze complex systems.

4. What are some practical applications of linear algebra?

Linear algebra has many practical applications, such as in computer graphics, cryptography, and data analysis. It is also used in engineering and physics to model and solve real-world problems.

5. How can I improve my understanding of these mathematical topics?

To improve your understanding of linear, multilinear, category, and graph theory, it is important to practice solving problems and working with these concepts. You can also read textbooks and attend lectures or seminars to deepen your understanding. Collaborating with other mathematicians and discussing these topics can also help improve your understanding and develop new insights.

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