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- Thread starter Sorgen
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- #2

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thanks!

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The only problem with something like shilov (or any book that isn't used by your professor), is the topics probably won't be in the same order as in your book, so if you work through it at the same time you may be working on two different things fairly consistently, so it will take more time. Just a thought.

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Get yourself the cheapest copy you can find. I definitely recommend a more theoretical book like Axler to read at the same time, or later.

I second the Axler recommendation. I'd also recommend Shilov and Friedberg.

I second shilov.

Thanks for the responses. I looked at these two books on amazon and they're pretty inexpensive, so I'm considering picking these up. The only issue that I have is that these two have gotten a lot of reviews that say that they're bad to learn linear algebra from. With no prior experience to linear algebra (besides maybe the course I'm about to take), do you think I would be able to tackle these books? Would it be better for me to push through the intro course and then try to take on those books?

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If you look at the websites of university math departments, you will see that the vast, vast majority of introductory linear algebra classes use texts that give a more gentle introduction --- Anton, Larson, or Strang.

I would buy one or two of those (older, used editions for no more than $10 each) to use as a supplement to your assigned text, for a first course. If you go on in math, you will get all the theory and proofs you want in upper division courses.

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Also, if you are a mathematics student you will eventually have to take rigorous and difficult courses, there's just no way of getting around that. Linear algebra is, in my opinion, one of the best subjects to be introduced to rigorous maths because many of it's proofs are in fact relatively simple, compared to say calculus where the proofs can be quite difficult to grasp.

Furthermore, rigorous maths books do not really get any simpler than Axler, they do, but not by much. If I am wrong, please provide evidence to the contrary.

Thanks for the responses. I looked at these two books on amazon and they're pretty inexpensive, so I'm considering picking these up. The only issue that I have is that these two have gotten a lot of reviews that say that they're bad to learn linear algebra from. With no prior experience to linear algebra (besides maybe the course I'm about to take), do you think I would be able to tackle these books? Would it be better for me to push through the intro course and then try to take on those books?

I don't really know what to say about these negative reviews. What I can tell you is that Axler is used at Harvard:

http://www.math.harvard.edu/~ctm/home/text/class/harvard/55a/08/html/hw.html

You can see here all the universities that use the book:

http://linear.axler.net/LADRAdoptions.html

That list includes Cornell, MIT, the various UofC universities, many state universities, Duke, etc.

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In this forum, you get a lot of responses from people who, frankly, can't relate to an average person. They are going out of their way to try to help you, but they think everyone is as gifted as they are, and that everyone taking a math class should concentrate on theory and proofs, and it's simply not the case. It's kind of like Peyton Manning coaching junior high football, and telling a 13-year old to throw the ball 60 yards down the field.

If you look at the websites of university math departments, you will see that the vast, vast majority of introductory linear algebra classes use texts that give a more gentle introduction --- Anton, Larson, or Strang.

I'd consider myself an average--if not below average--math student (my talent is more centered in English and the like). I survive because of my enthusiasm for the subject, not because of any real innate ability, and I've found that math is doable at pretty much any level of experience given sufficient motivation. Plus, Sorgen said that he or she is a pure math major, so learning linear algebra (an accessible area of math to begin with) from a theoretical perspective will be immensely useful down the road.

Thanks for the responses. I looked at these two books on amazon and they're pretty inexpensive, so I'm considering picking these up. The only issue that I have is that these two have gotten a lot of reviews that say that they're bad to learn linear algebra from. With no prior experience to linear algebra (besides maybe the course I'm about to take), do you think I would be able to tackle these books? Would it be better for me to push through the intro course and then try to take on those books?

Shilov's is a book that will serve you more as a reference down the road than as a first look. Axler, on the other hand, is a very good book from which to learn because it's entirely self-contained and is very accessibly written. If you've not had any proof-based courses before, it won't be EASY, but it's definitely manageable and it'll prepare you wonderfully for future math courses. If you feel moderately driven, go for it.

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I'd consider myself an average--if not below average--math student (my talent is more centered in English and the like). I survive because of my enthusiasm for the subject, not because of any real innate ability, and I've found that math is doable at pretty much any level of experience given sufficient motivation. Plus, Sorgen said that he or she is a pure math major, so learning linear algebra (an accessible area of math to begin with) from a theoretical perspective will be immensely useful down the road.

Shilov's is a book that will serve you more as a reference down the road than as a first look. Axler, on the other hand, is a very good book from which to learn because it's entirely self-contained and is very accessibly written. If you've not had any proof-based courses before, it won't be EASY, but it's definitely manageable and it'll prepare you wonderfully for future math courses. If you feel moderately driven, go for it.

Alright! I'll give it a go then. Thanks everyone for the help

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Hoffman/Kunze & Friedberg are the hardest books in the world until you learn how to deal

with a proof. If this is what you need then watch http://nptel.iitm.ac.in/video.php?subjectId=106106094 first. Lay's Analysis With

an Introduction to Proof & Discrete Mathematics Demystified are also great for first learning

proofs, they were for me anyway.

My advice is to use the http://nptel.iitm.ac.in/video.php?subjectId=122107036, module 2 onwards, along with the

2nd edition of Serge Lang's Linear Algebra (I really do mean the 2nd edition as it is

a combination of both of his linear algebra books & more) and also Golden Linear

Algebra. The reason I recommend the Golden book is the wealth of problems that

make explicit what the other books gloss over in terms of elementary calculations.

If you can get through this then use Friedberg, Hoffman/Kunze & the Handbook of

Linear Algebra to clear everything up.

with a proof. If this is what you need then watch http://nptel.iitm.ac.in/video.php?subjectId=106106094 first. Lay's Analysis With

an Introduction to Proof & Discrete Mathematics Demystified are also great for first learning

proofs, they were for me anyway.

My advice is to use the http://nptel.iitm.ac.in/video.php?subjectId=122107036, module 2 onwards, along with the

2nd edition of Serge Lang's Linear Algebra (I really do mean the 2nd edition as it is

a combination of both of his linear algebra books & more) and also Golden Linear

Algebra. The reason I recommend the Golden book is the wealth of problems that

make explicit what the other books gloss over in terms of elementary calculations.

If you can get through this then use Friedberg, Hoffman/Kunze & the Handbook of

Linear Algebra to clear everything up.

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- #13

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Well, brocks, Sorgen will already be using Lay, so what is wrong with having a book that is a bit more difficult to supplement it?

It's not a question of right or wrong; it's a question of helpful. If a student understands everything in the assigned text, then he may want to look at a more advanced text. But if he is having trouble with the assigned text, he may want to look at one that is less terse and has a gentler approach.

Also, if you are a mathematics student you will eventually have to take rigorous and difficult courses, there's just no way of getting around that.

But that was exactly my point. Getting to rigor eventually, as opposed to right off the bat, is the best path for most students. Even at MIT or CalTech, the only freshmen who take the honors courses that stress theory and proofs are those who have already taken an easier version in high school.

Linear algebra is, in my opinion, one of the best subjects to be introduced to rigorous maths because many of it's proofs are in fact relatively simple, compared to say calculus where the proofs can be quite difficult to grasp.

That opinion is almost universal, including among the authors of the texts I recommended. They *introduce* rigor at a pace that has proven to work with the majority of students, rather than only the most gifted.

To make a general comment, not necessarily focusing on the OP of this thread, there is no question about Shilov, or Spivak, or Landau being excellent texts, but I doubt there are very many people, even among their biggest fans, who can honestly say that those texts were their first exposure to LA or calculus or mechanics. Either they are extremely gifted, or they forget that they discovered how great Spivak or Landau were AFTER they had already been introduced to calculus or physics with a less rigorous text.

My opinion, for what it's worth, is that less harm can come from a learning curve that it too easy than one that is too difficult. The worst that can happen if someone takes the gentler path is that he might take an extra year to thoroughly master the subject, but he may get discouraged and abandon the subject altogether if forced along too fast. I readily acknowledge that opposing opinions exist, and are at least as valid and informed as my own.

- #14

hotvette

Homework Helper

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http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/

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It's not a question of right or wrong; it's a question of helpful. If a student understands everything in the assigned text, then he may want to look at a more advanced text. But if he is having trouble with the assigned text, he may want to look at one that is less terse and has a gentler approach.

Right. So Lay is simple and Axler is a bit harder. Or maybe if he has trouble with a text he needs to work a little harder. Like Chaostamer said, it's not about God-given talent, it's about how hard you work.

But that was exactly my point. Getting to rigor eventually, as opposed to right off the bat, is the best path for most students. Even at MIT or CalTech, the only freshmen who take the honors courses that stress theory and proofs are those who have already taken an easier version in high school.

You'd have to back that with fact. You don't need to have seen everything at a simple, proof-less level before you read a rigorous proof-based book. That would make everything twice as long. You only need to know how to do proofs.

That opinion is almost universal, including among the authors of the texts I recommended. They *introduce* rigor at a pace that has proven to work with the majority of students, rather than only the most gifted.

Um, no. Many universities cover honours calculus, i.e. rigorous calculus before linear algebra, or concurrently. Some even do number theory or something else first.

To make a general comment, not necessarily focusing on the OP of this thread, there is no question about Shilov, or Spivak, or Landau being excellent texts, but I doubt there are very many people, even among their biggest fans, who can honestly say that those texts were their first exposure to LA or calculus or mechanics. Either they are extremely gifted, or they forget that they discovered how great Spivak or Landau were AFTER they had already been introduced to calculus or physics with a less rigorous text.

My opinion, for what it's worth, is that less harm can come from a learning curve that it too easy than one that is too difficult. The worst that can happen if someone takes the gentler path is that he might take an extra year to thoroughly master the subject, but he may get discouraged and abandon the subject altogether if forced along too fast. I readily acknowledge that opposing opinions exist, and are at least as valid and informed as my own.

Landau is definitely a graduate level book. Spivak, I assume you mean his calculus book, requires only as a backround that you know something about how to do proofs, no previous exposure to calculus is needed. Anyway, I've had this debate so many times on this forum I just couldn't be bothered any more.

Mathematics is difficult. If you don't like that find another field to study.

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