21–30 of 62 posts
Earlier quoted context omitted.
> What makes LADR superior to this book? Axler's approach for determinants.
The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering linear functions from 2^n to 3^n? There's no geometric interpretation I'm aware of for this case; indeed, I doubt there is one, due to the inherent nature of finite fields.
Say what?
Earlier quoted context omitted.
The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering linear functions from 2^n to 3^n? There's no geometric interpretation I'm aware of for this case; indeed, I doubt there is one, due to the inherent nature of finite fields.
> linear functions from 2^n to 3^n Say what?
[1] https://archive.org/details/LinearAlgebraAndMatrixTheory
Earlier quoted context omitted.
> What makes LADR superior to this book? Axler's approach for determinants.
The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering linear functions from 2^n to 3^n? There's no geometric interpretation I'm aware of for this case; indeed, I doubt there is one, due to the inherent nature of finite fields.
Earlier quoted context omitted.
> What makes LADR superior to this book? Axler's approach for determinants.
People who are going to be users of applied linear algebra would probably benefit from a traditional treatment of determinants. Axler's book is excellent for people who plan to go for more pure mathematics in their studies.
What is nice about this book is it includes solutions to exercises. I wish more proof-theoretic books included exercises in the back of the book like this one does. Yes, you can prove something in multiple ways, and no that isn't a good excuse to not include at least one version of the proof in the back of the book. It really benefits the self-learner. The math community seems to have a lot of artificial gatekeeping…
In my experience (math teacher at a community college for 20 years and former Ph.D. student who quit before finishing the thesis) the overwhelming majority of people can not learn mathematics properly from a textbook. They need a teacher or someone to help with problems. It's very difficult to get to a point of really understanding the definition/motivation behind concepts. Even with a teacher it's hard to understand…
While you are right, a related issue is that most maths textbooks are atrocious at most aspects that aren't writing pages of equations. There is almost nobody sitting at the 3-way intersection of great mathematician, great writer and great educator who can then write great textbooks.
I've found books with terms like "History" and "Philosophy" in the title are much better places to learn about mathmatics; combined with wikipedia for formulas and details. Any book with excercises but no solutions or historical context has turned out to be basically useless to me, even as a reference (wikipedia is usually better for simple stuff).
I've got something like 4 books on statistics on my shelf at the moment. The only one that I've actually manged to read and learn something from has been Chatterjee's Philosophy of Statistics, because it talks about what techniques were developed in context of which problem, failed alternative approaches, explains what was confusing to some of the greatest minds in the history of statistics, etc. This has been vastly more useful in setting up a framework for what the world of statistics looks like that I can attach a whole bunch of proofs and suchlike too. It has been enlightening in a way that textbooks can't really manage.
What is nice about this book is it includes solutions to exercises. I wish more proof-theoretic books included exercises in the back of the book like this one does. Yes, you can prove something in multiple ways, and no that isn't a good excuse to not include at least one version of the proof in the back of the book. It really benefits the self-learner. The math community seems to have a lot of artificial gatekeeping…
In my experience (math teacher at a community college for 20 years and former Ph.D. student who quit before finishing the thesis) the overwhelming majority of people can not learn mathematics properly from a textbook. They need a teacher or someone to help with problems. It's very difficult to get to a point of really understanding the definition/motivation behind concepts. Even with a teacher it's hard to understand…
There's a difference between a person starting off in mathematics, versus someone who has done several undergrad/grad level courses and knows the "system". If you've gone through math formally earlier in your life, you should be able to pick up a math textbook and learn the subject. Unfortunately, many math textbook do not make this easy.
What is nice about this book is it includes solutions to exercises. I wish more proof-theoretic books included exercises in the back of the book like this one does. Yes, you can prove something in multiple ways, and no that isn't a good excuse to not include at least one version of the proof in the back of the book. It really benefits the self-learner. The math community seems to have a lot of artificial gatekeeping…
Earlier quoted context omitted.
What are your particular critiques about this book? There's nothing inherently inferior about a free resource. What makes LADR superior to this book? Finding a teacher isn't at all mandatory, you can do math yourself by completing exercises, checking your work, and looking at the solution afterwards. There are plenty of places to ask questions if you get stuck.
> What makes LADR superior to this book? Axler's approach for determinants.
Axler defines determinant as (up to a sign) the constant term of the characteristic polynomial, and he needs two different definitions for characteristic polynomial, one over R and one over C. Now what if the ground field is something else? Do we need yet another definition of characteristic polynomial in order to define the determinant? What if you are doing linear algebra over a commutative ring?
LADR actually presents a very narrow view about linear algebra : it treats linear algebra merely as finite-dimensional functional analysis. The readers can be hit hard when they need to do other (computational or theoretical) stuffs. Similar concerns had been voiced on the internet before. In particular, I think Darij Grinberg's comments (below the answer https://mathoverflow.net/a/16996) on LADR are rather spot on.
It's fine if you find LADR helpful. The book does have its merits (I like its clear and fluent writing and its neat proofs), but it has also its own shares of problems and there are other nice choices of books in the wild.
Earlier quoted context omitted.
> What makes LADR superior to this book? Axler's approach for determinants.
The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering linear functions from 2^n to 3^n? There's no geometric interpretation I'm aware of for this case; indeed, I doubt there is one, due to the inherent nature of finite fields.
This is probably the correct generalization of the volume definition.