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I always have to mention my all-time favorite introductory book on this subject:

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.




This, a thousand times this.

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'm curious - did you work through this book on your own, or as part of a class or with help from others?

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.


Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can.

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.


> Mathematicians are chronically lost and confused. It's our natural state of being, and it's okay to keep going and take what insights you can.

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.


Someone should tell us that sooner then. I've been trying to get over math anxiety for a long time, and having professors that say "it's simple, what don't you understand?" when you try to get help didn't help much.


A common joke among mathematicians.

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."


I started reading it for a class but I wasn't assigned all the problems. Later, I managed to propel myself through it by building confidence with the problems of the first few chapters (which were easy by that point) and then embracing the sunk-cost fallacy to get through the rest.

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.)


I actually skimmed the book over before I took my first course in linear algebra (doing some of the exercises on my own, but certainly not all--I skipped anything that seemed too hard or just too boring during this reading). At that point, there were definitely parts of the book that were challenging to me, and I just jumped over them.

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.


I agree. I am also working through this book and feel the same as you. I think also part of the problem maybe not being able to discuss this with other people. I'm trying very hard to get through it but I think I miss being able to discuss problems with other students in a traditional classroom setting. These sorts of discusses can really help clarify things rather than just staring at the page for a long time.


A side note: how do you feel when you've lost your motivation? Is it common to barely be able to keep your eyes open? Or is that just me!


It's not just you. It's probably easier to accept that I'm sleepy and should take a rest, than to acknowledge that I need to focus and think harder about the material.


Not just you. I'm procrastinating on one of those too-damn-hard math problems right now!


Oh yes, +1. Linear algebra is about vector spaces; matrices are a separate topic.

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.


No, he defines the determinant as the product of the eigenvalues, and this is almost the entire point for his writing the book!


This book treats linear algebra like any other mature mathematical subject. It just so happens that it's easier to understand than most mature mathematical subjects. But yeah this is just how any pure mathematician would approach the subject (which is why I liked it so much).


That particular series contains some of the best math texts I've read. My instructor used the Abbot text ( http://www.amazon.com/Understanding-Analysis-Undergraduate-T... ) for our real analysis course. The books are very aptly priced and are extremely well written.


What is the equivalent book for other topics in math (I'm trying to self study) e.g. Calculus, Abstract Algebra, Probability, Statistics?


Any idea how this compares to Hoffman-Kunze? I liked Hoffman-Kunze because, despite being dense, its no-nonsense and very rigorous.


I do so agree. Axler's book is also by far the most Geometric book on Linear algebra that I've ever read.




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