
Linear Algebra and Applications: An Inquiry-Based Approach - henning
https://scholarworks.gvsu.edu/cgi/viewcontent.cgi?article=1021&context=books
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jcranmer
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.

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tonyarkles
I went through EE and CS. EE we started using matrices exactly how you
describe it: here’s a system of linear equations, here’s how you write them in
matrix form, here’s how you invert them to solve the original system. Turn the
crank, answer pops out. I had my trusty HP49G, and I could solve linear
systems all day.

Then in CS I took a computer graphics course and it was rotation and
translation matrices all day every day.

Then there was a digital communications course where we touched on orthogonal
basis functions, and some matrix voodoo related to that and how to get
orthogonal vectors out of the mess.

And then finally I took the required CS linear algebra course offered by the
math department, where we started from scratch. Here’s a vector (psh, I know
vectors!), here’s a vector space (hmmm this is new), and building the rest of
it up from there. I _really_ wish that had come earlier on, but I was very
very happy to finally have a bit of a theoretical understanding of how these
tools I’d been using actually worked.

~~~
jammygit
I feel like my university only taught calculation, not theory, when it came to
linear algebra. It’s like the equivalent of a “12 hacks to rotate a matrix”
article. The theoretical books I find however give no explanation for the
definitions etc, ie, WHY are the dot/cross products defined the way they are.
It’s as though they feel matrixes are natural phenomenon that you should just
memorize the properties of, which is also nonsense.

The entire field is defined by such terrible books. I’d love to be wrong
though if somebody has a recommendation

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faizshah
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
Thank you for the recommendation.

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Vaslo
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.

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anjc
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

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ch
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.

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

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melvinroest
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

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marktangotango
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!

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josinalvo
for this use case, I use favorite rather than the upvote

