Axler or Shilov for Linear Algebra?

In summary, the conversation revolves around choosing a book for self-learning linear algebra and the syllabus for an introductory algebra course. The two books being considered are "Axler" and "Shilov" and it is suggested that "Shilov" would be more suitable for the syllabus. The conversation also mentions the topics covered in the algebra syllabus, such as group theory, ring theory, and field theory. Various resources are recommended for studying these topics, including the notes of the speaker's class and a book by Mike Artin.
  • #1
EmSeeSquared
29
1
Hi all,

So I'm going to have my first exposure to linear algebra and I've completed Calc 1 and 2.
I've seen Axler and Shilov numerous times and I'm having a hard time choosing it.

Here's my syllabus for my Linear Algebra Course.
Matrices, Gauss reduction, invertibility. Vector spaces, linear independence, spans, bases, row space, column space, null space. Linear maps. Eigenvectors and eigenvalues with applications. Inner product spaces, orthogonality.

For Linear Algebra which one would be best suited for my syllabus? Axler or Shilov. The order in which the content is represented does not matter to me since I'm planning to self learn the whole thing before the classes start in 3-4 months. :)

Now After Linear Algebra, I will have an Introductory Algebra course. The syllabus is below(Please suggest some good books on it as well):

Further linear algebra: equivalence relations, the quotient of a vector space, the homomorphism theorem for vector spaces, direct sums, projections, nilpotent linear transformations, invariant subspaces, change of basis, the minimum polynomial of a linear transformation, unique factorization for polynomials, the primary decomposition theorem, the Cayley-Hamilton theorem, diagonalization. Group theory: subgroups, cosets, Lagrange's theorem, normal subgroups, quotients, homomorphisms of groups, abelian groups, cyclic groups, symmetric groups, dihedral groups, group actions, Caley's Theorem, Sylow's Theorems. Ring theory: rings, subrings, ideals, quotients, homomorphisms of rings, commutative rings with identity, integral domains, the ring of integers modulo n, polynomial rings, Euclidean domains, unique factorization domains. Field theory: subfields, constructions, finite fields, vector spaces over finite fields.

Thanks
 
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  • #2
I think Axler is a second book on linear algebra whereas Shilov is an introduction. It will also cover the topics on further linear algebra in your algebra syllabus. That material on group theory and ring theory is in most abstract algebra books. I like Mike Artin's Algebra but there are easier books. My own class notes are free on this page, e.g. the notes for 843-1 are on basic group theory through Sylow.

http://www.math.uga.edu/~roy/
 
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Likes muraii and EmSeeSquared
  • #3
mathwonk said:
I think Axler is a second book on linear algebra whereas Shilov is an introduction. It will also cover the topics on further linear algebra in your algebra syllabus. That material on group theory and ring theory is in most abstract algebra books. I like Mike Artin's Algebra but there are easier books. My own class notes are free on this page, e.g. the notes for 843-1 are on basic group theory through Sylow.

http://www.math.uga.edu/~roy/

Thanks :) I'll get Shilov then. And look into your notes. :) Hopefully I'll complete LA and AC during my holidays and next year I can do RA and IA while revising LA and AC :)
 

FAQ: Axler or Shilov for Linear Algebra?

1. What is the difference between Axler's and Shilov's approach to teaching Linear Algebra?

Axler's approach focuses on the conceptual understanding of Linear Algebra, whereas Shilov's approach emphasizes on the computational aspect of the subject. Axler's book is more abstract and proof-based, while Shilov's book is more concrete and calculation-oriented.

2. Which book is better for beginners in Linear Algebra?

This ultimately depends on the learning style of the individual. Axler's book may be more challenging for beginners due to its abstract approach, but it provides a strong foundation for further studies in mathematics. Shilov's book may be easier for beginners to grasp, but it may not cover all the abstract concepts in depth.

3. Which book has more real-world applications?

Both books cover a wide range of real-world applications of Linear Algebra, but Shilov's book may have more examples and exercises related to engineering and physics, while Axler's book focuses more on theoretical applications.

4. Which book is more suitable for self-study?

Again, this depends on the individual's learning style and background knowledge. Axler's book may be more challenging for self-study due to its abstract approach, while Shilov's book may be more accessible for self-study as it provides more concrete examples and exercises.

5. Are there any major differences in the topics covered in both books?

Both books cover the core topics of Linear Algebra, such as vector spaces, matrices, and linear transformations. However, Axler's book may have a more modern and abstract approach, while Shilov's book may cover some additional topics such as eigenvalues and eigenvectors in more depth.

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