Thoughts on Linear Algebra with Applications / 5e Bretscher

[matthew9]

Hey y'all! I am taking my first linear algebra course next semester and we are using Bretscher's book. I have heard some pretty awful things about the way it is laid out, so I would like to hear your thoughts on the text. For some background information I have completed Calculus 1-3 (using Stewart) and am a Mathematics major, but I would prefer to have a clear textbook since this is a new subject for me.

 

[gleem]

I can't say anything about Bretscher's book however if you want one that is devoted to applications to physics, an old Monograph, that I used in my Linear Algebra course, by J. Headings "Matrix Theory for Physicists" was found to be useful. It covers applications to geometry, mechanics, electromagnetic theory, quantum mechanics and special relativity.

 

[verty]

I personally find matrices to be quite daunting. For example, the complex numbers can be represented by matrices and the quarternions can be represented by matrices. So matrices seem to be quite powerful and general.

And... a vector is a matrix as well. And surely any rectangular subset of a matrix is a matrix, even a single cell. So matrices are these amorphous things and we need to reign them in somehow.

And linear algebra would seem to be about matrices and matrix operations. For example, we have the equation |A - λI| = 0 which is an algebraic equation. So there definitely is an algebraic point of view. And I actually like this, that an eigenvalue is a solution to that formula.

As the OP is a math major, I want to recommend a book with this point of view but that will also be challenging (I always think a book should be challenging.) I therefore recommend https://www.amazon.com/dp/B00CWR4Y9M/?tag=pfamazon01-20 as it seems to have this algebraic approach.

 

[vanhees71]

Vectors, particularly for a math major, are way more than matrices of numbers. I'd thus not recommend such a text to a mathematician. On the other hand I don't know many English math textbooks, because I learned all my math from German textbooks :-(.

 

[Fredrik]

I would prefer to have a clear textbook since this is a new subject for me.

Then I suggest that you use Treil either as a supplement or as your main text.

 

[matthew9]

I personally find matrices to be quite daunting. For example, the complex numbers can be represented by matrices and the quarternions can be represented by matrices. So matrices seem to be quite powerful and general.

And... a vector is a matrix as well. And surely any rectangular subset of a matrix is a matrix, even a single cell. So matrices are these amorphous things and we need to reign them in somehow.

And linear algebra would seem to be about matrices and matrix operations. For example, we have the equation |A - λI| = 0 which is an algebraic equation. So there definitely is an algebraic point of view. And I actually like this, that an eigenvalue is a solution to that formula.

As the OP is a math major, I want to recommend a book with this point of view but that will also be challenging (I always think a book should be challenging.) I therefore recommend https://www.amazon.com/dp/B00CWR4Y9M/?tag=pfamazon01-20 as it seems to have this algebraic approach.

I actually do have the Stoll book and I find it to be quite good, but I haven't had the time to dive into it as much as I would have liked by now. The main issue I will be facing is that we are going to have homework from the Bretscher text and if his examples aren't clear, it seems to reason that his homework questions will be equally as vague (at least that is my experience with textbooks). I trust that the lectures from my professor will give me the pertinent information, but it is always good to have a reliable textbook for a truly rigorous understanding. Thanks for the advice!

 

[matthew9]

Then I suggest that you use Treil either as a supplement or as your main text.

I really appreciate the link to the Treil book! I will absolutely keep it in mind for when I start next semester. Thank you.

 

[verty]

Vectors, particularly for a math major, are way more than matrices of numbers. I'd thus not recommend such a text to a mathematician.

I don't know, for a first course it should be okay.

 

[verty]

I actually do have the Stoll book and I find it to be quite good, but I haven't had the time to dive into it as much as I would have liked by now. The main issue I will be facing is that we are going to have homework from the Bretscher text and if his examples aren't clear, it seems to reason that his homework questions will be equally as vague (at least that is my experience with textbooks). I trust that the lectures from my professor will give me the pertinent information, but it is always good to have a reliable textbook for a truly rigorous understanding. Thanks for the advice!

Well hopefully that won't be a problem because you'll have a good understanding of how the things are built up. And I suppose one good thing is, you can collaborate with other people in your class and you may have insights that they don't have and vice versa. So I'm sure it'll work out just fine.

Vanhees did make a good point but for a first course it should be okay to not have to think in terms of axioms.