6 ms·

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-Undergrad

by nilkn 13y ago

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-Mathematics/dp/0387982582 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.

shriphani 13y ago
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-Texts-Mathematics/dp/0387950605 http://www.amazon.com/Understanding-Analysis-Undergraduate-T... ) for our real analysis course. The books are very aptly priced and are extremely well written.
jjoonathan 13y ago
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.
j2kun 13y ago
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).
rahimnathwani 13y ago
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.
- j2kun 13y ago
  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.
  - vsbuffalo 13y ago
    > 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.
    - epaladin 13y ago
      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.
enupten 13y ago
I do so agree. Axler's book is also by far the most Geometric book on Linear algebra that I've ever read.
akater 13y ago
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 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.
- j2kun 13y ago
  No, he defines the determinant as the product of the eigenvalues, and this is almost the entire point for his writing the book!
abhgh 13y ago
Any idea how this compares to Hoffman-Kunze? I liked Hoffman-Kunze because, despite being dense, its no-nonsense and very rigorous.
billspreston 13y ago
What is the equivalent book for other topics in math (I'm trying to self study) e.g. Calculus, Abstract Algebra, Probability, Statistics?

by nilkn 13y ago

shriphani 13y ago

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-Texts-Mathematics/dp/0387950605 http://www.amazon.com/Understanding-Analysis-Undergraduate-T... ) for our real analysis course. The books are very aptly priced and are extremely well written.

jjoonathan 13y ago

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.

j2kun 13y ago

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).

rahimnathwani 13y ago

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.

j2kun 13y ago
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.
- vsbuffalo 13y ago
  > 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.
  - epaladin 13y ago
    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.

enupten 13y ago

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

akater 13y ago

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 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.

j2kun 13y ago
No, he defines the determinant as the product of the eigenvalues, and this is almost the entire point for his writing the book!

abhgh 13y ago

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

billspreston 13y ago

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