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Saturday, December 19, 2009

Linear Algebra ( Books for 2010 )

Professor Gross ( Harvard ) said in one of the abstract algebra ( e-222 ) lectures that " you can't learn too much linear algebra ". I always liked linear algebra but I thought I knew most of it. I could not have been more ignorant. I discovered a book today 'Advanced Linear Algebra' by Steven Roman, a book from the Springer Graduate Texts in Mathematics series. It contains two parts, part 1: basic linear algebra contains ten chapters called:
- 1. Vector spaces
- 2. Linear transformations
- 3. Isomorphism theorems
- 4. Modules I
- 5. Modules II
- 6. Modules over a PID
- 7. The structure of a linear operator
- 8. Eigenvalues and eigenvectors
- 9. Real and complex inner product spaces
- 10. Structure theory for normal operators
Part 2: topics, contains another 9 chapters.
I think I am ready for part 1, since M208 contains linear algebra as well, this book definitely comes on my 2010 list. I just decided I am going to make a list for the math books I want to study in 2010 besides M208, MT365.

I was actually studying a book on Group Representation Theory. That's a topic which relies heavily on linear algebra. When I was studying Maschke's Theorem I realized I had to review my linear algebra, especially inner product spaces. And then I found Roman's book. My understanding of vector spaces (1) is ok, linear transformations (2) as well, and if not, M208 has lots of stuff on that. I studied the isomorphism theorems (3) in group theory, I think they are more or less the same in linear algebra. I have some notion about modules (4,5,6): like vector spaces but with scalars from a ring instead of from a field. Have to study them deeper, I suppose. Maybe the CG Modules ( vector spaces where the vectors can be multiplied with group elements as well as scalars ) I studied are a sort of modules, have to check it out. Looking forward to learn more about structure of linear operators (7) and (10) as well. The stuff in Eigenvalues and eigenvectors
(8) and Real and complex inner product spaces (9) is familiar but most likely goes much deeper here.

From a first browse through the book I can say that I like the style. Clearly written and enough examples. I hate books that don't have examples. I think that authors who don't include examples in their books are - A) too lazy, or B) not really willing to communicate their knowledge, or C) sadistic. In all cases bad people. - The book is packed with exercises but alas for the self-study student: no answers. I haven't made my mind up about authors who do provide exercises but keep the answers to themselves. Fortunately I found some problem books on advanced linear algebra as well. More on that another time, perhaps.

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