What are good books for a third course in Linear Algebra?

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  • #1
fxdung
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What are the suitable books in linear algebra for third course for self-study after reading Linear Algebra done right by Axler and Algebra by Artin?
 
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  • #2
S.G. Janssens
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Maybe you would like Roman's Advanced Linear Algebra.
 
  • #3
fxdung
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But Roman omit some proofs. Is there any book more detail?
 
  • #4
Sam Gallagher
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What subject are you interested in? Ideas might be:
* Module theory
* Multilinear algebra
* Functional analysis
* Representation theory
* Numerical methods in linear algebra
If you've come this far, you'll know the field of 'linear algebra' is really a first-step towards each of the above, and they all offer different extensions. If you can be more specific people could recommend better books?
 
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  • #5
fxdung
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I haven't had that knowledge. So I need a general book higher than Axler's book
 
  • #6
MidgetDwarf
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Why make repeated threads? You asked about second course linear algebra books about a month ago. You are telling me you finished Artin and Axler in a month? Read what you have...
 
  • #7
fxdung
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I have finished Axler, and I am reading Artin.I ask the question preparing for the time I finish Artin.
 
  • #8
Demystifier
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What are the suitable books in linear algebra for third course for self-study after reading Linear Algebra done right by Axler and Algebra by Artin?
Are you sure that you want a third book on linear algebra? Maybe you just want more advanced stuff on abstract algebra, not necessarily linear.
 
  • #9
fxdung
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I need to go deeper in Linear Algebra
 
  • #10
S.G. Janssens
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I need to go deeper in Linear Algebra
Need or want?

Both is fine, linear algebra is alive as a research field by itself, not only as background knowledge for other fields. (Although its connection with other fields makes it arguably more interesting.)

Maybe try browsing and reading some articles. This is freely accessible:

https://journals.uwyo.edu/index.php/ela

zbMATH curates a searchable journal list that is free to consult.
 
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  • #11
StoneTemplePython
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I haven't had that knowledge. So I need a general book higher than Axler's book
You are asking for a book after Artin and Artin has two very nice chapters on Module Theory and Representation theory respectively so this comment doesn't make much sense.

Btw, if you want more comprehensive knowledge an easy upgrade is to do the (especially starred) problems in Artin 1st ed. He dumbed down the problems in the 2nd edition -- i.e. cut down the raw number and eliminated a lot of high insight but difficult problems.
 
  • #12
Infrared
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I need to go deeper in Linear Algebra

Why? Are there specific topics that you want to learn about?
 
  • #13
martinbn
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Kostrikin and Manin "Linear algebra and geometry".
 
  • #14
fresh_42
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I have finished Axler, and I am reading Artin.I ask the question preparing for the time I finish Artin.
... in which case it shouldn't be a problem to answer ...
Why? Are there specific topics that you want to learn about?
... because ...
I need to go deeper in Linear Algebra
... requires specific topics that you call deeper, in order to make specific suggestions.
 
  • #15
fxdung
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Some online University teach PhD on Linear Algebra, so I need a deeper in general Linear Algebra
 
  • #16
fresh_42
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Let me see where you are. Maybe this helps us to figure out what you should do.

Let ##\psi## be a linear transformation of the inner product space ##E##. Define the linear automorphism ##\exp \psi## by
$$
\exp\psi =\varphi (1)
$$
where ##\varphi (t)## is the family of linear automorphisms defined by
$$
\dot\varphi (t)=\psi\circ \varphi (t)\, , \,\varphi (0)=\operatorname{id}.
$$
Prove that
$$
\varphi (t)=\exp(t\psi) \quad (-\infty <t<\infty ).
$$

https://www.amazon.com/dp/0387901108/?tag=pfamazon01-20
p. 258
 
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  • #17
fxdung
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I have not finished undergraduate in math(self-study).I have just only read Basic Analysis by Peterson Vol 1-3 and Linear Algebra done right by Axler and preparing to read Algebra by Artin.So I haven't been able to solve problem in Greub.Is it good to read Greub after finish Artin?
 
  • #18
MidgetDwarf
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I have not finished undergraduate in math(self-study).I have just only read Basic Analysis by Peterson Vol 1-3 and Linear Algebra done right by Axler and preparing to read Algebra by Artin.So I haven't been able to solve problem in Greub.Is it good to read Greub after finish Artin?
By read, do you mean working through the exercises without looking at solutions?
 
  • #19
fxdung
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I intent to do exercices after re-reading the books. I like have a general view about mathematics.
 
  • #20
MidgetDwarf
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I intent to do exercices after re-reading the books. I like have a general view about mathematics.
Thats not how science or math books works. You have to do the exercises...
 
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  • #21
Vanadium 50
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I intent to do exercices after re-reading the books.
This explains why you have posted so many messages struggling with various things. @MidgetDwarf is right.
 
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  • #22
fresh_42
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So I haven't been able to solve problem in Greub.Is it good to read Greub after finish Artin?
Sure, Greub is a good book. But it will not solve your problem. You can read a thousand books and still don't make a progress. Those books you already have are good books, too, and working through them should qualify you to solve most linear algebra problems.

The key attitude when reading a scientific book is: "I do not believe any of what is written! Book, convince me!"

E.g. a rotation is a linear transformation. Then you have to think that it is not, but the book or your personal work makes you see that it is one. What does it mean, that it is linear? Can I draw a picture? Why does it have those properties? Is it correct under any circumstances, or are there exceptions? These kinds of questions, a pencil and a lot of paper to scribble on must be present all the time when you read a textbook.
 
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  • #23
fxdung
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Yes,you are right! And I need every thing must be "demonstrated" .I have spirit "must be convinced by logic".But I want save time not to do exercices at first time.And in short time I have general view about Math.
Is that way OK?
 
  • #24
berkeman
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Yes,you are right! And I need every thing must be "demonstrated" .I have spirit "must be convinced by logic".But I want save time not to do exercices at first time.And in short time I have general view about Math.
Is that way OK?
IMO, No. If you work through a chapter in a textbook and skip the exercises, you are wasting your time (and ours, BTW) if you want to be sure you are effectively learning the material. The exercises are a test of your understanding. "Self Learning" does not mean that you get to skip the learning part and just get a general idea of the material.

How about we pause this thread until you go back and re-read your first couple of textbooks and work through the exercises. When you can show us the solution to the straightforward math quiz question posed by @fresh_42 we can resume this discussion thread...
 
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  • #25
Demystifier
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Yes,you are right! And I need every thing must be "demonstrated" .I have spirit "must be convinced by logic".But I want save time not to do exercices at first time.And in short time I have general view about Math.
Is that way OK?
I understand why others criticize you, but I have to tell you that I use similar strategy when I only want to get a big picture in a short time. After getting a big picture, then I can more easily motivate myself to study the details. So if that strategy works for you too (obviously, it doesn't work for everybody), go with it!
 
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  • #27
mathwonk
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For what it's worth, here is my summary of the theory of normal forms of finite dimensional linear operators (more or less the content of a first or second course of linear algebra):

Given a linear operator T on a finite dimensional k-vector space V,
V has a decomposition into a product of subspaces Wj on each of which T is equivalent to the action of multiplication by X on a quotient space k[X]/(fj) of a polynomial ring. I.e. if Wj is a subspace corresponding to a polynomial fj, there is an isomorphism from Wj to k[X]/(fj) under which the action of T on Wj corresponds to multiplication by X on k[X]/(fj). In particular, fj is the minimal polynomial of T on Wj. Thus understanding the behavior of T on V is accomplished by finding these polynomials fj and the corresponding subspaces Wj.

The first clue to finding these polynomials is that their product is equal to the "characteristic polynomial" of T, so the problem is to find the appropriate factorization of that polynomial. E.g. if the characteristic polynomial is irreducible over k, there is only one subspace W=V, and f = the characteristic polynomial. In general, the distinguished subspaces can be chosen so that the corresponding sequence of monic polynomials f1,...,fr successively divide each other, and when this is done this special sequence of polynomials, called "invariant factors" of T, is uniquely determined by T. In this case the largest degree one, fr, is the minimal polynomial of T on V.

Thus if the characteristic polynomial is a product of distinct irreducible polynomials, there is again only one subspace W=V, r=1, and f equals the characteristic polynomial. In general, the “invariant factor decomposition”, can be computed by hand from any matrix for T, by diagonalizing the associated "characteristic matrix", using the Euclidean algorithm in k[X].

Two operators S,T are “similar”, i.e. T = (U^-1)SU for some invertible operator U, if and only if S,T have the same invariant factors. If M is a matrix for S in some basis, another way to say T is similar to S, is that there is some basis in which T also has the matrix M.

A second standard decomposition exists where the polynomials fj in the model
spaces k[X]/(fj) are all powers of irreducible polynomials. For this decomposition, the
sequence of polynomials fj is almost uniquely determined by T, except for a chosen
ordering of the irreducible polynomials.

This second decomposition, called the “generalized Jordan decomposition”, always exists in theory, but can be computed in practice only for those examples where the irreducible factors of the characteristic polynomial of T can actually be found, e.g. for a “triangular” matrix.

A special case of the Jordan decomposition occurs precisely when the minimal
polynomial factors completely into distinct linear factors. Then the Jordan form,
which may or may not be effectively computable, is a diagonal matrix. This is
always the case when the matrix consists of real entries which are symmetric about
the main diagonal, although even then one may not be able to perform the factorization in practice, nor to actually find the numerical entries on the diagonal. In that event one may turn to approximation techniques to estimate these "eigenvalues".

If a real matrix does not equal its "transpose", (= its reflection about the diagonal), but does commute with it, then the minimal polynomial is again a product of distinct irreducible factors, hence of degree ≤ 2, and the subspaces Wj all have dimension ≤ 2. If a complex matrix commutes with the complex conjugate of its transpose, its minimal polynomial is also a product of distinct irreducible factors, and since these must all be linear it is actually diagonalizable. In all cases where the matrix either equals or commutes with its transpose, the decomposing subspaces can all be chosen mutually orthogonal , indeed subspaces corresponding to distinct irreducible polynomials are automatically orthogonal.
 
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