I can't understand most of the topics on my Linear Algebra Book.

[sarah22]

I don't know if the book or I'm the one is the problem .

Last day, I borrowed a book titled Linear Algebra, 2nd Edition - Kenneth Hoffmann And Ray Kunze. Then tried to read most of the topics there but didn't get most of it.

I studied Calculus by my own and understand most of it. But when I tried to study Linear Algebra, it really burned my brain up. Can anyone give me some tips on how to study this subject? Do you have any good book for a self-teach man like me? I'm studying this topic on my own because my university didn't offer this class but I really need it for game programming. That's the reason why I borrowed a book and will teach myself with it.

 

[aPhilosopher]

I would get a different book. I haven't looked at that book myself but from what I understand, it's for a second course in linear algebra that goes on to do things like work over an arbitrary field and such.

I first learned linear algebra from "Linear Algebra: An introduction to abstract mathematics" by Robert Valenza. If you study that book then you will be well grounded in the core theory of linear algebra. It omits some practical stuff, for instance it focuses on matrices as linear transformations and omits LU-decomposition and stuff like that but you're only going to be dealing with 4x4 matrices anyways.

If you can't find that book, just pick another book on linear algebra from a library or something. You can do better than the one that you have for an introduction from what I understand.

 

[VeeEight]

Two important things a beginner should learn are:
Solving systems of linear equations using matrix row reduction
Linear Transformations (basic concepts like range and null space, dimension theorem)
You can probably find info on the web (if I have time I'll try to find a good link). Read through examples and do problems until you are comfortable with the basic material. Once you have a general understanding of what is happening you can move onto move in depth stuff.

 

[Tac-Tics]

Start off with vectors in R^n. Know how to add and scale vectors.

Understand what a linear combination is. For example, the vector u = (4, 5, 0) can be written as a linear combination of i = (1, 0, 0) and j = (0, 1, 0) by saying u = 4 i + 5 j.

Know what a basis is. A basis is a set of vectors (such as i, j, k) such that any other vector is a unique linear combination of those vectors.

Understand what a linear transformation is. It's a vector-valued function f that takes a vector and satisfies f(au) = af(u) and f(u + v) = f(u) + f(v) for scalars a and vectors u, v. These things play very nicely with bases and linear combinations! If {i, j, k} is a basis, and I know f(i), f(j), and f(k) separately, then I know f(v) for ANY vector. It's amazing.

Because I only need to know f(i), f(j), and f(k) to fully define a linear transformation, we can represent the linear transformation with a matrix. The matrix's size will depend on the dimension of the domain and codomain of the linear transformation.

If a matrix is square, that means the domain and codomain are the same. So, for example, a 3x3 matrix takes vectors in R^3 to vectors in R^3. These are nice. Some square matrices have inverses. So a matrix that rotates a vector can be "undone" with its inverse by rotating it the opposite direction at the same angle. An important tool to invert a matrix is the determinant. It's interesting to see how matrix multiplication problems correspond to systems of linear equations with several unknowns.

That's my crash course. What parts of those are you familiar with? Which are giving you the most grief?

 

[VeeEight]

Know what a basis is. A basis is a set of vectors (such as i, j, k) such that any other vector is a unique linear combination of those vectors.

And the basis vectors are linearly independent

 

[jasonRF]

I would agree with others - get a different book. I have seen folks on these forums suggest Hoffman and Kunze for intro, but in my opinion that only makes sense for people who would be honors math majors. My favorite intro book is "elementary linear algebra" by Anton. I am familiar with the 7th edition, and it probably has everything you will need, and includes extra optional chapters on applications. You can get used copies from amazon for <$10. For an e-book, there is "Linear Algebra" by Jim Hefferon - you can find it on google. It is a little more advanced than Anton, but is free and has a free solutions manual as well.

Good luck!

Jason

 

[aPhilosopher]

My favorite intro book is "elementary linear algebra" by Anton.

A class I was in used Anton's Calculus book. I didn't buy it because homework didn't count for enough of the grade for me to care about doing it but the other students seemed to understand it very well the few times that I did go to class. As a general rule, good book implies good author implies more good books.

Edit, of course, we could have just had a good teacher. I didn't happen to think so though.

 

[symbolipoint]

I once had a used copy of "Elementary Linear Algebra" by Anton, and the Hefferon e-book and tried to use them. I could not get very far in either before becoming unable to continue dealing with them. Too Hard! I was not attending any course work at the time, so not sure if that made any difference. I wish I knew what my head is missing. I was fine with the material in some Intermediate Algebra books, but from a Linear Algebra book? No progress.

 

[aPhilosopher]

That's odd. I find algebra much more difficult than linear algebra. Linear algebra of finite dimensional vector spaces is easy if you get a good introduction. The elementary ones that I've looked at are incredibly dull, and hence difficult. Maybe it's just me, but the glossy pages and color pictures of people skiing or what have you on every other page just turn me off incredibly.

More to the point, they seriously over emphasize algorithms and don't spend nearly enough time developing concepts. It's just incredibly stupid and tedious. Matrix calculations are best done by a computer. A human needs to know how to read the result of one and how to set one up. If you understand what's going on though, you can't help but know how to do it. There's no need to drill it over and over again. Better to spend time writing proofs and hence learning the concepts. What's the point of knowing how to do a calculation if you don't know what calculation to do anyway?

I again highly recommend Valenza's book, "Linear Algebra: An introduction to Abstract Mathematics". If I recall correctly, he only assigns one computation in the whole book. He has you invert a matrix, just so you can say that you did. Every other exercise, and there are a lot of them, is about proving something. The proofs are easy too. I was able to do every one and I'm not that smart. It's actually the only good math book that I've been able to do that for the first time through other than the book that I learned calculus from. It really contains a solid introduction to how to prove things. It's a brilliant book.

 

[LogicalTime]

Prof Strang has a full course of linear algebra which includes video lectures up at MITOCW:
http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/CourseHome/index.htm

He speaks rather slowly which is frusterating unless you download the lectures and watch them at 1.5x using VLC. I made a page that shows how to do this:
http://sites.google.com/site/variablespeedlectures/

Good Luck!

 

[Landau]

And the basis vectors are linearly independent

That is already contained in the 'unique' part.