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Linear Algebra and Applications: An Inquiry-Based Approach

203 points by henning 7y ago

Vaslo 7y ago
The fact that it doesn’t start with some unreadable mathematical notation that is just the author trying to show how smart they are give me hope. Looks like a really good introductory source just skimming the first few chapters.
melvinroest 7y ago
I have no clue how this is having 52 votes and no comments on it. How am I supposed to know this is a good book? I'll highlight the goals of this book as it explains more about the title. > We place an emphasis on active learning and on developing students’ intuition through their investigation of examples. For us, active learning involves students – they are DOING something instead of being passive learners. I found this goal the most interesting. > To help students understand that mathematics is not done as it is often presented. We expect students to experiment through examples, make conjectures, and then refine their conjectures. We believe it is important for students to learn that definitions and theorems don’t pop up completely formed in the minds of most mathematicians, but are the result of much thought and work
- deleted 7y ago
  [deleted]
- davidivadavid 7y ago
  I've been thinking about something along similar lines for a while and was wondering if there were any math/physics/science textbooks that not only took a more historical approach to teaching their subject, but also had a more "problem based" motivation. In other words, I'm looking for textbooks that start from the problems scientists were trying to solve when they came up with their discoveries, instead of presenting the theory ex nihilo, and then retroengineering a purpose to that theory (which to my recollection is how 99% of textbooks work).
  - melvinroest 7y ago
    Oh! If you have any recommendations, I’d be interested :)
- marktangotango 7y ago
  The reason I upvoted was with intent to review later. I personally found that after two semesters of linear and a BS Mathematics I didn’t know jack about linear algebra. I came to the conclusion that I should’ve studied physics or engineering if I’d wanted to actually learn how to use it!
  - josinalvo 7y ago
    for this use case, I use favorite rather than the upvote
anjc 7y ago
I haven't gone through this yet but I really like the idea of each new concept being described in the context of a useful application. Thanks OP
ch 7y ago
Cool. I want to try and work through this text, just to assess how useful it is. The approach is an interesting one. Looks like this will have to become a weekly goal. Maybe one chapter a week? Seems possible.
faizshah 7y ago
Along these lines, my stats professor recommended a really nice book that offers a case studies based approach for grad level stats: http://www.statisticalsleuth.com/ http://www.statisticalsleuth.com/ I've been going through it by implementing the solutions in jupyter notebooks. They have the datasets and code in R so it's easy to work with and work out the solutions.
- dmitryminkovsky 7y ago
  Thank you for the recommendation.
jcranmer 7y ago
Pedagogically, the challenge to teaching linear algebra is that you start with "here's systems of linear equations, we can put them into matrices and now here's row operations to solve them," and you end up with "now matrices are actually representations of linear operators on vector spaces, let's analyze the properties of this specific operator." Usually, this is also coupled with a reluctance to actually discuss vector spaces, since the meat of it involves abstract algebra, which usually comes after linear algebra. Failing to tackle this challenge appropriately can leave students confused about properties that seem apparently random (trace and determinant are big offenders here), or textbooks bringing something up only to never mention it again (null space is often an example here). On top of this, there is also the multiple notation problem (admittedly, not as bad as calculus, where there are too many notations for derivative) and the minor issue that many of the algorithms taught in the book aren't used in practice because of numerical stability issues. It has been so long since I've taken linear algebra, and I've taken abstract algebra courses since then, that I can't really compare this book to the approach that I learned. Skimming the book, the thing that jumps out the most to me is that LU factorization and determinants are shoved surprisingly late in the book [1], and eigenvalues are "previewed" quite early. I'm not sure that's a good approach: LU factorization is important because backsolving the L and U matrices is more numerically stable (and sparser, when you're dealing with sparse matrices) than the inverse matrix, and it works even if your matrix isn't square. Furthermore, determinants tie in better to row operations, and their weird application with Cramer's rule is another way to solve a set of linear equations: you don't want to introduce Cramer's rule months after you finished treating matrices as stepping stones to solving linear equations. The book does cover vector spaces, although in a bit of a dance around not covering abstract algebra. I'm not sure it's an effective introduction of vector spaces, although it could well suffice to ease the pedagogical trap mentioned earlier. On the other hand, if it's going to dive that far into vector spaces, it would probably be helpful to have some more sections on matrices over fields that aren't real numbers (i.e., complex numbers (make sure to mention conjugate transpose and Hermitian matrices!), rational numbers, and finite fields). [1] Strassen's algorithm for matrix multiplication is described before LU factorization, to give you an idea of how weird the ordering ends up being.
- jimhefferon 7y ago
zhamisen 7y ago
Hopefully this topology book will be in the same style: https://bookstore.ams.org/text-58 https://bookstore.ams.org/text-58

203 points by henning 7y ago

The fact that it doesn’t start with some unreadable mathematical notation that is just the author trying to show how smart they are give me hope. Looks like a really good introductory source just skimming the first few chapters.

melvinroest 7y ago

I have no clue how this is having 52 votes and no comments on it. How am I supposed to know this is a good book? I'll highlight the goals of this book as it explains more about the title. > We place an emphasis on active learning and on developing students’ intuition through their investigation of examples. For us, active learning involves students – they are DOING something instead of being passive learners. I found this goal the most interesting. > To help students understand that mathematics is not done as it is often presented. We expect students to experiment through examples, make conjectures, and then refine their conjectures. We believe it is important for students to learn that definitions and theorems don’t pop up completely formed in the minds of most mathematicians, but are the result of much thought and work

deleted 7y ago
[deleted]
davidivadavid 7y ago
I've been thinking about something along similar lines for a while and was wondering if there were any math/physics/science textbooks that not only took a more historical approach to teaching their subject, but also had a more "problem based" motivation. In other words, I'm looking for textbooks that start from the problems scientists were trying to solve when they came up with their discoveries, instead of presenting the theory ex nihilo, and then retroengineering a purpose to that theory (which to my recollection is how 99% of textbooks work).
- melvinroest 7y ago
  Oh! If you have any recommendations, I’d be interested :)
marktangotango 7y ago
The reason I upvoted was with intent to review later. I personally found that after two semesters of linear and a BS Mathematics I didn’t know jack about linear algebra. I came to the conclusion that I should’ve studied physics or engineering if I’d wanted to actually learn how to use it!
- josinalvo 7y ago
  for this use case, I use favorite rather than the upvote

anjc 7y ago

I haven't gone through this yet but I really like the idea of each new concept being described in the context of a useful application. Thanks OP

ch 7y ago

Cool. I want to try and work through this text, just to assess how useful it is. The approach is an interesting one. Looks like this will have to become a weekly goal. Maybe one chapter a week? Seems possible.

faizshah 7y ago

Along these lines, my stats professor recommended a really nice book that offers a case studies based approach for grad level stats: http://www.statisticalsleuth.com/ http://www.statisticalsleuth.com/ I've been going through it by implementing the solutions in jupyter notebooks. They have the datasets and code in R so it's easy to work with and work out the solutions.

dmitryminkovsky 7y ago
Thank you for the recommendation.

jcranmer 7y ago

Pedagogically, the challenge to teaching linear algebra is that you start with "here's systems of linear equations, we can put them into matrices and now here's row operations to solve them," and you end up with "now matrices are actually representations of linear operators on vector spaces, let's analyze the properties of this specific operator." Usually, this is also coupled with a reluctance to actually discuss vector spaces, since the meat of it involves abstract algebra, which usually comes after linear algebra. Failing to tackle this challenge appropriately can leave students confused about properties that seem apparently random (trace and determinant are big offenders here), or textbooks bringing something up only to never mention it again (null space is often an example here). On top of this, there is also the multiple notation problem (admittedly, not as bad as calculus, where there are too many notations for derivative) and the minor issue that many of the algorithms taught in the book aren't used in practice because of numerical stability issues. It has been so long since I've taken linear algebra, and I've taken abstract algebra courses since then, that I can't really compare this book to the approach that I learned. Skimming the book, the thing that jumps out the most to me is that LU factorization and determinants are shoved surprisingly late in the book [1], and eigenvalues are "previewed" quite early. I'm not sure that's a good approach: LU factorization is important because backsolving the L and U matrices is more numerically stable (and sparser, when you're dealing with sparse matrices) than the inverse matrix, and it works even if your matrix isn't square. Furthermore, determinants tie in better to row operations, and their weird application with Cramer's rule is another way to solve a set of linear equations: you don't want to introduce Cramer's rule months after you finished treating matrices as stepping stones to solving linear equations. The book does cover vector spaces, although in a bit of a dance around not covering abstract algebra. I'm not sure it's an effective introduction of vector spaces, although it could well suffice to ease the pedagogical trap mentioned earlier. On the other hand, if it's going to dive that far into vector spaces, it would probably be helpful to have some more sections on matrices over fields that aren't real numbers (i.e., complex numbers (make sure to mention conjugate transpose and Hermitian matrices!), rational numbers, and finite fields). [1] Strassen's algorithm for matrix multiplication is described before LU factorization, to give you an idea of how weird the ordering ends up being.

jimhefferon 7y ago

zhamisen 7y ago

Hopefully this topology book will be in the same style: https://bookstore.ams.org/text-58 https://bookstore.ams.org/text-58