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I found Linear Algebra Done Right by Axler to be a fantastic theoretical text that makes everything click together from day 1. It's derived from his paper Down

by nextos 12y ago

I found Linear Algebra Done Right by Axler to be a fantastic theoretical text that makes everything click together from day 1. It's derived from his paper Down with Determinants. He starts by explaining that linear algebra is about studying the properties of linear transformations over vector spaces, and then goes on to explain what this means. In many theoretical books it takes ages to get this message, and in some practical ones you get lost in a sea of matrices without even understanding what a matrix actually is. Furthermore all his proofs are very slick.

jjoonathan 12y ago
Seconded. I was very skeptical of LADR when I was forced to use it freshman year. I thought I already knew all the practical stuff from Strang's book, but learning linear algebra from the mathy angle has paid off again and again and again. Choices of units & coordinate systems are impossibly confusing without having a foundation in how to reason independently of such arbitrary choices and add them in later. I think the turning point came when I realized that "abstract" linear algebra was actually much closer to physical reality than "applied" linear algebra. Superficially, it seems like the exact opposite state of affairs from computer engineering, where grids of numbers take you closer to "bare metal" and "higher level reasoning" takes you further away. In math, it's the grids of numbers that are further removed from "bare metal" and the abstract reasoning that you can count on when all else fails! I didn't need to be convinced that I would eventually come to regret having an insecure foundation so much as I was mixed up about up vs down and trying to build a foundation in the middle of the sky.
mafribe 12y ago
In my experience, Linear Algebra Done Right by Axler is a great second book on linear algebra. But it makes sense to most students only if they've seen and done concrete matrix calculations before. Giving it to a mathematical novice is likely to lead to baffelment. Too abstract.
- nextos 12y ago
  I don't think it's too abstract. It contains plenty of geometrical intuition. And it makes things very clear and he says them very _early_. E.g., a matrix represents a linear transformation. It took me ages to understand this at high school because I was lost in a sea of matrices.
nilkn 12y ago
I can't recommend this book more. I read through about half of it before I took my first real course on linear algebra. The result? I got literally 100% in that class -- I didn't miss a single point on anything the entire semester. The teacher was flabbergasted, but I had my secret weapon -- Axler. The rest of the class was relying on the course textbook, some awful fat thing that I'm pretty sure nobody could ever really learn from. Axler's book made the concepts so crystal clear to me that it felt as if I'd known them since birth. I can't guarantee that everyone is going to have the same experience, but yeah, it's a good book.
- eli_gottlieb 12y ago
  Wow, really? I mean, I got an A- in linear algebra with some normal old textbook I don't even remember, and certainly learned it all in terms of linear transformations over vector spaces... but then again, I didn't have it as a doubled class with differential equations in the first place. Is Axler that good?
  - nilkn 12y ago
    It was partially a fluke I'm sure. I'm notoriously sloppy with arithmetic and no book could save me from that. It's not necessarily that Axler is that good. It's that it's just solidly good all throughout. It has the right focus. The goal of the book was very clearly to explain linear algebra, not to make money or get contract deals with college bookstores. And the proofs are very clearly written. I'm sure there are other books out there which are just as good as Axler. But compared to the more popular linear algebra textbooks, at least at the time, it seemed like it was in a class of its own.
- nextos 12y ago
  Any analysis recommendations? I have Hubbard & Hubbard in mind.
  - nilkn 12y ago
    I don't have any particularly noteworthy recommendations for ordinary analysis or real analysis, but for complex analysis I did find that Needham's Visual Complex Analysis was just as revolutionary for me as Linear Algebra Done Right (if not more so).
tptacek 12y ago
Anyone want to study group this book? I did Strang a few years back, along with the OCW lectures, but it wasn't very proofs-centric.
- pakled_engineer 12y ago
  MIT has some more proof centric courses http://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-fall-2013/ http://ocw.mit.edu/courses/mathematics/18-700-linear-algebra...
- nextos 12y ago
  I'd be quite interested to keep in touch with some people studying these topics, even if we don't follow the same pace or curriculum. Despite having studied a decent dose of math during my undergrad (and currently during my grad), I keep on finding awesome books like Axler, Hubbard, Halmos, Neuveu... that make revisiting math so much more illuminating. Everything clicks much more quickly and I never loose momentum now. Some of these books I found online, so I'd definitely like to exchange more references, problems, tips & experiences.

by nextos 12y ago

jjoonathan 12y ago

Seconded. I was very skeptical of LADR when I was forced to use it freshman year. I thought I already knew all the practical stuff from Strang's book, but learning linear algebra from the mathy angle has paid off again and again and again. Choices of units & coordinate systems are impossibly confusing without having a foundation in how to reason independently of such arbitrary choices and add them in later. I think the turning point came when I realized that "abstract" linear algebra was actually much closer to physical reality than "applied" linear algebra. Superficially, it seems like the exact opposite state of affairs from computer engineering, where grids of numbers take you closer to "bare metal" and "higher level reasoning" takes you further away. In math, it's the grids of numbers that are further removed from "bare metal" and the abstract reasoning that you can count on when all else fails! I didn't need to be convinced that I would eventually come to regret having an insecure foundation so much as I was mixed up about up vs down and trying to build a foundation in the middle of the sky.

mafribe 12y ago

In my experience, Linear Algebra Done Right by Axler is a great second book on linear algebra. But it makes sense to most students only if they've seen and done concrete matrix calculations before. Giving it to a mathematical novice is likely to lead to baffelment. Too abstract.

nextos 12y ago
I don't think it's too abstract. It contains plenty of geometrical intuition. And it makes things very clear and he says them very _early_. E.g., a matrix represents a linear transformation. It took me ages to understand this at high school because I was lost in a sea of matrices.

nilkn 12y ago

I can't recommend this book more. I read through about half of it before I took my first real course on linear algebra. The result? I got literally 100% in that class -- I didn't miss a single point on anything the entire semester. The teacher was flabbergasted, but I had my secret weapon -- Axler. The rest of the class was relying on the course textbook, some awful fat thing that I'm pretty sure nobody could ever really learn from. Axler's book made the concepts so crystal clear to me that it felt as if I'd known them since birth. I can't guarantee that everyone is going to have the same experience, but yeah, it's a good book.

eli_gottlieb 12y ago
Wow, really? I mean, I got an A- in linear algebra with some normal old textbook I don't even remember, and certainly learned it all in terms of linear transformations over vector spaces... but then again, I didn't have it as a doubled class with differential equations in the first place. Is Axler that good?
- nilkn 12y ago
  It was partially a fluke I'm sure. I'm notoriously sloppy with arithmetic and no book could save me from that. It's not necessarily that Axler is that good. It's that it's just solidly good all throughout. It has the right focus. The goal of the book was very clearly to explain linear algebra, not to make money or get contract deals with college bookstores. And the proofs are very clearly written. I'm sure there are other books out there which are just as good as Axler. But compared to the more popular linear algebra textbooks, at least at the time, it seemed like it was in a class of its own.
nextos 12y ago
Any analysis recommendations? I have Hubbard & Hubbard in mind.
- nilkn 12y ago
  I don't have any particularly noteworthy recommendations for ordinary analysis or real analysis, but for complex analysis I did find that Needham's Visual Complex Analysis was just as revolutionary for me as Linear Algebra Done Right (if not more so).

tptacek 12y ago

Anyone want to study group this book? I did Strang a few years back, along with the OCW lectures, but it wasn't very proofs-centric.

pakled_engineer 12y ago
MIT has some more proof centric courses http://ocw.mit.edu/courses/mathematics/18-700-linear-algebra-fall-2013/ http://ocw.mit.edu/courses/mathematics/18-700-linear-algebra...
nextos 12y ago
I'd be quite interested to keep in touch with some people studying these topics, even if we don't follow the same pace or curriculum. Despite having studied a decent dose of math during my undergrad (and currently during my grad), I keep on finding awesome books like Axler, Hubbard, Halmos, Neuveu... that make revisiting math so much more illuminating. Everything clicks much more quickly and I never loose momentum now. Some of these books I found online, so I'd definitely like to exchange more references, problems, tips & experiences.