Liner Algebra Done Right (http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat...)
Its breakthrough is its focus on the basic algebraic properties of vector spaces and linear maps between them. It de-emphasizes matrix computations and especially determinants (they are covered, but only insomuch as they are necessary).
In my experience, the result of a typical linear algebra course is most students don't fully understand the determinant and more importantly they don't understand the proofs of major theorems which involve long manipulations of the determinant. They also don't understand the more algebraic side of the subject because they aren't given a chance to--it's not covered in much detail. The result is they don't understand the subject overall much at all.
This book is based on the observation that the abstract algebra involved in linear algebra is actually remarkably easy, much more so than arcane determinant manipulations.
My first introduction to Linear Algebra came from Strang's OpenCourseWare videos and, a year later, a class that used his book. Despite having gone through the material twice, I struggled to apply it to actual problems. Connecting the real world (err, models of the real world) to the blocks of numbers was the sticking point. What are the matrix's units? How do I write down a matrix that does what I want? I looked things up and prayed that whatever black magic the mathematicians on wikipedia had invoked in their derivations would match the implicit assumptions I knew I had to be making.
Then I was forced to re-take linear algebra in college. We used Axler's book and suddenly everything made sense. The blocks of numbers that you type into matlab are simply shadows of the mathematical concepts of a vector and linear transformation. The abstract mathematical concepts are actually closer to the real world concepts they model because they don't have to depend on arbitrary choices of coordinate system, etc. Once you understand how to model real-world quantities with abstract vectors and linear maps (easy) and how to translate those coordinate-free equations into coordinate-dependant matrix equations (easy), THEN you understand the material enough to apply it without fumbling around trying to paste other peoples' formulas together (hard).
OP's book looks well written but it entirely follows the tradition of emphasizing the blocks of numbers over the abstract concepts. Linear maps don't appear until page 75, while blocks of numbers enter on page 5! That's nuts! Use Axler.
I started reading it, attempting most of the exercises at the end of each chapter, and some of the 'as you should verify' parts. I found the 'aha' moments during the text or when solving the exercises really enjoyable, but I got stuck often enough that I began to only pick up the book when I felt super-alert.
I stopped about a third of the way through, a few pages before the introduction of eigenvectors and eigenvalues. I would like to pick it up again, but am worried about how to maintain motivation the next time I'm faced with a page where I'm stuck/confused for 30 minutes.
EDIT: Thank you to the people who replied. Your empathy (~"you're not alone in getting stuck") and encouragement (~"you don't have to grok everything the first time through") have given me new motivation to try this again.
I suggest you don't worry too much about verifying every claim and doing every exercise before moving on. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. It's healthy, if you get stuck on such a problem, to think about other problems or come back to them later. It's not uncommon to find that by the time you revisit them you've literally grown so much (mathematically) that they're trivial.
This needs to be emphasized more in learning maths. Once I got over this fact and was guided by my interest rather than wanting to grok everything, I actually started understanding a lot more.
Two mathematicians are working on a problem and one makes a claim to which the other replies, "That's trivial!" The pair stare at the claim for ten minutes. The other says, "Is it trivial?" They go home, think about it all night, publish three papers about it over the next two years, and then they agree, "Yes, it's trivial."
One problem stood out. I forget the chapter, but I'll never forget the problem: prove that normed vector spaces are inner product spaces IFF the parallelogram law holds. It took about half a day to crack that problem and I was completely lost for about 90% of it. (That's not a strategy. The first paragraph was my strategy. My only point here is that you aren't alone in getting stuck.)
Then when I took the actual course, I more or less ignored the main course textbook and instead tried to tie whatever happened in the last lecture with what I found in Axler.
I considered Axler my secret weapon. It honestly felt like I had access to secret insights that trivialized the class and granted me the same intuition the professor had.
Unfortunately, Axler does not define determinant as a volume scaling coefficient and goes the still arcane way of “(-1)^\deg…” which does not really do justice to the concept.
Check out Sergei Winitzki's “Linear algebra via Exterior Products” https://sites.google.com/site/winitzki/linalg It's free for download and treats determinants in a more geometrically inclined way. BTW, it's written by a physicist, not a mathematician.
I'm hoping for a Math textbook written like its an abusive drill sergeant.
"You think matrix inversion is hard, huh? We're still just laying the mothafuckin groundwork, you halfwit. You ain't BEGUN to see hard yet.... You have no CONCEPTION of what a hard math problem might look like. This math is not hard. This math is easy. And I'm gonna make you bust your brain against these exercises until you make it look easy. Now pick up that goddam pencil and WORK your brain."
I'm a teacher. My colleague is a former marine sergeant with a Computer Science degree. We'd soon make a ma(thematicia)n out of you.
Currently, PhD in statistics and nearly a qualifed actuary (I've done all the maths bits of the qualification).
You are doing Spaghetti Monster's work.
P.S: I'm in a top CS Grad school, so no, thank you, for your offer of consdescension. I'm sure, you and your man-ly pals will have a happy time making more soldiers than mathematicians.
After searching for the solution, I found this awesome resource that combines linear algebra, differential equations and graph theory to model salt flows in various networks of tanks.
My point is that the linked article uses linear algebra as part of a problem statement that requires more than linear algebra for its analysis.
Thanks for all your comments!
This is the followup book from the No bullshit guide to math and physics and is written in the same style. I'm still doing some final touchups on the prerequisites chapters (Ch1 and Ch2) and I want to beef-up the Applications chapter, but the core material is done.
Several of my students asked to see the book---even unfinished---because they wanted to use it to study for their finals so I decided to make a pre-release. Once I finish the Applications chapter I'll create the print version and a nicer looking eBook (sans-serif). The current version is intended for easy printing at home of sections that you have to study for.
@Irishsteve thanks for posting! I am on a plane right now going to SF (tourism mostly, but if anyone wants to talk textbook business... hit me up) and I couldn't resist the $10 for 1h of internet time. I had a nice surprize when I checked HN!
Hence the current ordering of chapters...
Any material that is placed in a textbook to inflate the page count and legitimize the $100+ price of the book.
Actually, the forces of bullshit are strongest in freshman calculus and physics texts; linear algebra books are much better on that front.
aybe I should make it more prominent.... but usually for UGRDs they focus on eigen-decomp so I put the main focus there too...
It looks theoretical and very deep - not what you need for passing an undergraduate math course. Maybe exactly what you need to pass a graduate level statistics course.
Note: The author should add a: "I may want this." field with an email subscription option :/.
For the eBook, I'm targeting tablets mostly
(see http://cnd.mcgill.ca/~ivan/miniref/noBSmath_preview.pdf ), but this is not ideal for smaller screens.
I have a kindle forma as a work in progress, but it's not idea as some equations really need a wider screen, so not sure how to break them up...
Long term, I'll have to figure out how to generate clean epub with math as images (I would prefer MathJax, but not all ebook readers support js). That should fix the small screen format.
Thx for your feedback!
Do you know of any epubs with math in them that I could imitate?
Would you be willing to try out some of my .epub testing and give me feedback? If so please send me an email.
Well then the title worked! The idea is to disguise a serious textbook that teaches the material and understanding, but not make it look like a non-serious text so as not to scare off the reader.