# Quick question on intro to linear algebra book

1. Sep 5, 2016

I'm looking at purchasing Algebra (2nd Edition) by Michael Artin, is this a good book to purchase as my first intro to linear algebra book for self learning?

2. Sep 5, 2016

### micromass

No, it's not. Artin's book barely touches linear algebra. Sure, it does determinants and vector spaces and all that jazz. But you should get a book that focuses solely on linear algebra.

Furthermore, if you never heard of linear algebra before, then you should get an easier book.

3. Sep 5, 2016

oo micromass! we meet again old friend.. I got that book from the MIT site under the intro to algebra class syllabus. I'm not sure what an easier/good book is for intro to linear algebra tho. Any alternative suggestions?

4. Sep 5, 2016

### micromass

Artin is an excellent book which you definitely should go through once. But right now, I don't think it's a very good idea to do it.

What's your current knowledge of linear algebra? For example, do you know what a matrix is? Do you know how to multiply and invert matrices? Do you know what a determinant is?

5. Sep 5, 2016

Nothing, I've scene what matrix's look like and done some small multiplying/inversion but it didnt count for anything. Something from ground zero.

6. Sep 5, 2016

### micromass

OK, what about other mathematics? Stuff like analytic geometry: equations of lines and planes? Dot product? Trigonometry?

7. Sep 5, 2016

I finished an introduction to calculus and vectors. I'm currently working on
http://ocw.mit.edu/resources/res-18-006-calculus-revisited-single-variable-calculus-fall-2010/ to help me prepare for a more advanced calculus book. The book i'm using with this is Calculus and Analytic Geometry (9th Edition) by Thomas and Finney
https://www.amazon.com/Calculus-Analytic-Geometry-George-Thomas/dp/0201531747

You've helped me before micromass, this has a better summary of where I was and the calculus course I linked is what i'm taking to prepare for what I was referring to on this post

https://www.physicsforums.com/threa...c-feedback-on-math-book-to-learn-next.878148/

Last edited by a moderator: May 8, 2017
8. Sep 5, 2016

### micromass

Last edited by a moderator: May 8, 2017
9. Sep 5, 2016

Sounds good, ordered both of them. Couple weeks and i'll have some more work to do. heh

Thanks again for your help!

10. Sep 5, 2016

### micromass

Don't order the books immediately! You should always read the first few sections to see whether you even like the book. It's not because I like a book that it is suitable for you. Books are very personal things.

11. Sep 5, 2016

I got the practice problems, i'll read a bit into the book and see how it feels. I have a couple sample pages to try out.

12. Sep 5, 2016

### smodak

Last edited by a moderator: May 8, 2017
13. Sep 5, 2016

I'm not a big fan of khanacademy for anything more than the practice questions they have. I liked his explanations when I first started math, after I got exposed to it a bit more I find his rambles a lot and confused me more at times than needed.

I was going to watch these lectures
http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/
I think I saw one from yale as well which should be good stuff. I'll take a look at Hello Again, Linear Algebra, see what it's all about.

Thanks a lot! :D

14. Sep 6, 2016

### ibkev

I like Pavel Grinfeld's stuff too, on youtube he has additional lectures for tensor calculus to go with his other book as well.
What I especially like about youtube for watching lectures is that if you find any lecture too slow moving you can speed it up to 1.25x or 1.5x :)

Last edited by a moderator: May 8, 2017
15. Sep 15, 2016

### mathwonk

Crash course in linear algebra:
Once you understand the concept of dimension of a vector space, and in particular know what a basis is, the next concept is that of a linear transformation, and then the fundamental problem in linear algebra is to classify all linear transformations of a finite dimensional space to itself. The basic model example is the transformation defined by multiplication by X, on the quotient space k[X]/(f) where f = a0+a1X+...+an-1X^(n-1) + X^n is a monic polynomial of degree n. This space has natural basis 1,X,X^2,...,X^(n-1), and in this basis the given map permutes the basis vectors cyclically (i.e. 1 is sent to X, X is sent to X^2,...) until the last one X^(n-1), which is sent to X^n, which in the quotient space equals the following linear combination of the basis vectors: -a0 - a1X -a2X^2-....-an-1X^(n-1). The fundamental theorem says that every linear transformation of a finite dimensional space is a product of these models. I.e. this is essentially the most general linear transformation.

A refinement says that we can reduce to the case where the polynomial f is a power of an irreducible polynomial. Hence over an algebraically closed field like the complex numbers, we can assume the polynomial is (X-c)^n, where c is a constant. In this case, it is more natural to take as basis the set 1, (X-c),...,(X-c)^(n-1), i.e. powers of (X-c) instead of powers of X. In this basis the map sends each vector (except the last) to the next basis vector plus c times itself, and sends the last vector just to c times itself. In this basis the map is said to be in "Jordan form".

The simplest case is when the power of (X-c) is one, and the map is a product of copies of the map multiplication by X on the space k[X]/(X-c), i.e. the space is a product of subspaces on each of which the map is just multiplication by a constant. These are called diagonalizable maps, and there exist some important criteria to recognize some such maps, the simplest of which, over the real numbers, is that the matrix for the map is symmetric about the main diagonal. These theorems are called "spectral theorems".

I would also benefit from some clarification by micromass of his statement that artin barely touches on linear algebra, since he devotes chapters 1,3,4,7 to elementary linear algebra, then combines it with group theory in chapters 8 and 9, (representations of linear groups), and finally in chapter 12, applies module theory to deduce the jordan normal form. So in a sense he gives a complete treatment of all the linear algebra I outlined above, and even uses linear algebra to motivate and illustrate group theory.

I would guess that the message is that Artin's treatment is an advanced one and not recommended for an introduction. It was apparently intended as one for MIT students however and does cover pretty much everything, if somewhat efficiently.

Last edited: Sep 17, 2016