I used Jim Hefferon's Linear Algebra to get back into the subject a few years ago. Great balance of rigorous and intuitive. Free book available here http://joshua.smcvt.edu/linalg.html/
Ok, so Hefferon was my textbook for 2 full semesters. Lots of problems with this book. Hefferon is an algebraist, so the computational aspect of linear algebra ie. the stuff you are most likely to use linalg for at your workplace, has been given short shrift. Instead, the author gives fullblown treatment to theorem proving in vector spaces. So you are spending a lot of time constructing isomorphisms between subspaces....that shit is neat for a math major, which I was, but now that I'm in the workplace, I find what the world calls "linear algebra" isn't what Hefferon cares about.
eg. One of the basics you must absolutely know, or so I've been told, is how to do a PCA, and why a PCA is not the same as regression. So if you've got multidimensional data, say data in 7 dimensions ( risk ratio, fico score, salary, age etc etc ), you want to know if you can reduce that whole set to say 1 dimension. ie. can you transform the original set of 7 variable vectors into a much smaller set of 1 variable vectors that captures 90% of the information in the original ? If so, how ? Turns out you construct a linear combination of correlated variables. Ok, but there's infinitely many combinations. So how ? Well, construct it in such a fashion that the variance of the combination is maximized. That component is called the Principal Component, and its the single most used technique to reduce dimensionality in the industry. So you've taken a 7-dimension space and nicely reduced it to a 1-dimension subspace with maximum variance that explains 90% of the original 7 dimension space. Its quite amazing actually.
http://en.wikipedia.org/wiki/Principal_component_analysis
Similarly, there are a whole bunch of very common stuff you'd actually use linear algebra for in CG - translation, rotation in 3D, shear matrices and the like. Hefferon doesn't do those either.
In portfolio theory which is my bread and butter (http://en.wikipedia.org/wiki/Portfolio_theory ) almost everything I do is very properly linear algebra. But again, Hefferon doesn't offer any insight into how/why finding a vector with minimum variance can yield better returns in a vector space of portfolios.
Your problem with the book seems to be that it is general, rather than applicable to your particular field.
Why on earth would expect that it would be? I took and enjoyed linear algebra using that text-- it didn't specifically address the QM I deal with as a physicist, but since the author can't predict each student's future, I'd hardly characterise that as a problem with the text!
If it was used as a course on portfolios, ok, bad choice -- but directed against a general linear algebra text, your criticism doesn't seem very valid.
>> rather than applicable to your particular field.
No, I'm saying Hefferon is not applicable to any field. He eschews application in favor of formal theorem proving in vector spaces, in isolation of any real-life examples. That is great, but that is not the typical use case of anybody out there who actually uses linear algebra in industry, whether in computer graphics or portfolio theory or multivariate statistics or what have you.
Just to clarify for the crowd, portfolio theory, at least as far as its connection to linear algebra, is the mathematics of a covariance matrix. You are talking about different ways to decompose such a matrix. There are many ways and the justification for any of these methods goes beyond linear algebra and is really a special area of finance theory more than linear algebra.
Long before reading the book, I spent a lot of time with financial modeling of all kinds, but your complaint doesn't resonate with me at all. I went on to study machine learning and found the ideas in Hefferon laid a good foundation. It might be fair to point out that is not a cookbook and you don't use it to learn how to wrangle LAPACK, but I think it's over the top to say what you just did.
Being "not the typical use case... in industry" doesn't make it bad. It just makes it a pure mathematics book. Formal theorem proving in vector spaces is the typical use case of a pure mathematician; this book is for them.
If you're looking to do application, use an applied book. Gilbert Strang's "Linear Algebra and its Applications" is quite good.
I'll second that. Strang's book is by far my favorite introductory math book. He addresses fundamentals quite well, and gives good treatment to applications such as linear programming and game theory.
