
Down with Determinants (1995) - sdenton4
http://www.axler.net/DwD.html
======
CalChris
Axler seems to have won that battle. His textbook _Linear Algebra Done Right_
is widely used at 308 universities including Berkeley, Stanford and MIT. He
has a PDF available without proofs, videos, etc. _3blue1brown_ likes the book.

[http://linear.axler.net/](http://linear.axler.net/)

I suffered through determinants.

~~~
nextos
Axler's book is fantastic. But sadly Springer altered the typesetting on the
3rd edition. A really classic and clear LaTeX layout got turned into something
much less clear. This freaked me out. Look inside and compare:

* Second edition: [https://www.amazon.com/dp/0387982582](https://www.amazon.com/dp/0387982582)

* Third edition: [https://www.amazon.com/dp/3319307657](https://www.amazon.com/dp/3319307657)

I wonder whether widespread adoption of his book pushed editors to make it
look flashier and watered down. The contents are the same though.

~~~
mathgenius
Wow this is tragic. I'm guessing it serves whatever market that Springer has
identified. But I'm not sure that's such a good thing: at some point the more
details you add to the exposition the less clear it becomes. The reader needs
to stand on their own two feet, especially in mathematics. Some people seem to
be good at memorizing endless rules and details, so I can see this serving
those people. But those are the people that can just follow the "determinants
path" that this book was originally meant to disavow. Sigh.

~~~
DennisP
Doesn't seem _that_ bad, I just checked inside both and the content does look
identical other than the visual style and some examples added in the 3rd.

------
omazurov
Determinants have a very basic intuition behind them: it's the stretch factor
of the n-volume of a linearly transformed unit n-cube (area of a linearly
transformed unit square in 2D, volume of a linearly transformed unit cube in
3D, etc.) Why would one want to banish them from linear algebra?

~~~
whatshisface
The author alludes to that by pointing out that they're needed for jacobians,
which are essentially also stretch factors.

However, a nice geometric interpretation does not nice math make. If you
remember their _definition_ , the one with the sub-products and the
alternating +/-es, and then imagine trying to prove that it _has_ that
geometric interpretation - you've arrived at a huge pain for undergrad math
students that are just being introduced to matrices.

~~~
jessriedel
> then imagine trying to prove that it has that geometric interpretation

Exactly. We leaned heavily on determinants in my freshman linear algebra class
but I went at least an additional year before I even heard the interpretation,
much less could prove it from the standard terrible definition.

------
hexane360
If anyone's looking for a (high level) overview of linear algebra, I'd highly
recommend 3blue1brown's video series:
[https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQ...](https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)

It's mostly graphical, and is really helpful in forming and cementing an
intuition for linear algebra.

~~~
InclinedPlane
I really hate the way that mathematics is taught in most school systems, it's
all machinery with none of the beauty and underlying understanding. I was
lucky enough to maintain an interest in mathematics sufficient to allow me to
basically teach myself, with a heavy reliance on an underlying conceptual
understanding. A lot of things like calculus and linear algebra are mostly
just about understanding the basic concepts and then building up some
experience working with them. There's no need to memorize gazillions of
formulae or anything like that, you just need to actually _know_ what you're
doing when you do the work, but you will inevitably rely on tables and
references except for very simple work.

Which is to say, I could not recommend 3blue1brown's videos more highly, they
are an invaluable aid to learning linear algebra and actually helping you
understand what is you're doing when you're doing these various operations to
"solve problems".

------
experiment0
This is a great paper.

As a counterpoint, one place where determinants are incredibly useful is in
Hartree-Fock theory, where they effective encode the Pauli exclusion principle
(or anti-symmetry requirements) of atomic orbitals.

[https://en.wikipedia.org/wiki/Hartree–Fock_method](https://en.wikipedia.org/wiki/Hartree–Fock_method)

~~~
nategri
Also: Cross products.

~~~
Koshkin
One problem with cross product, it only exists in three dimensions.

~~~
wolfgke
No, it exists in dimension 3 and 7:

> [https://en.wikipedia.org/wiki/Seven-
> dimensional_cross_produc...](https://en.wikipedia.org/wiki/Seven-
> dimensional_cross_product)

EDIT: Nore precisely: A common way to axiomatize the cross product yields a
cross products exactly in dimension 3 and 7.

------
tzahola
I had the same uneasy feeling about determinants when I was studying linear
algebra at the university. Years later I found Sheldon Axler’s “Linear Algebra
done right”, and I loved it!

------
CamperBob2
Maybe I need a little more handholding than the average linear algebra
student, but if the language in that paper made any sense to me at all, I
probably wouldn’t need any instruction on determinants.

~~~
zodiac
Unfortunately I think the paper is written for an audience of other math
lecturers, to convince them to not use determinants in their own classes, and
not for beginning linear algebra students

------
pacala
> A complex number λ is called an eigenvalue of T if T −λI is not injective.

Uhm, what the intuition behind _that_?

~~~
sgdpk
You can think of it in these two steps:

1\. λI stretches every vector (the whole space, really) by a factor λ.

2\. Saying that the function is not injective means you lose information: when
you apply it on some object and get a result, you can't trace back what was
the original object, as there may be several. (There is no inverse function,
then). In linear algebra, this only happens because there is some direction of
space where all the vectors get collapsed to zero.

In short, T-λI collapses some line of vectors to zero.

So, when you took the effect of λI from T, you make it a lossy transformation
in some direction. This means that _in that direction_ T had the effect of
stretching all vectors by a factor of λ.

You gain some geometric understanding of T.

It is sort of intuitive, but the language may obscure it a little if you are
not used to it.

~~~
Myrmornis
Thanks!

> So, when you took the effect of λI from T,

If I understand right, you’re saying that there’s an interpretation in terms
of the geometry of the T transformation, of subtracting this diagonal matrix
from T. Multiplication of matrices is composition of transformations, I get
that, but I’m not so sure what adddition/subtraction is.

~~~
sgdpk
Yes, that's right. Addition is just applying the transformations separately to
the same vector and adding the result. So what this is saying is that if you
apply λI to a vector in that particular direction, then there is nothing left
to add to get the effect of T.

Ideally you would like to do this for all n directions of space, and that way
you completely describe what T does in simpler terms: it just stretches things
differently in different directions. It's not always possible though. The
matrices that allow this are called diagonalizable and the process of finding
the stretch factors (eigenvalues) is called diagonalization.

Just a caveat: if an eigenvalue is complex, the effect is not as simple as a
stretch, but the interpretation is very similar.

~~~
Myrmornis
Thanks very much. I'm glad I asked. Clearly, I had failed to really
internalize what it means to be a linear operator!

------
forkandwait
Axlers book is lovely, but my (amateur) opinion is that determinants are
pretty damn intuitive and useful in the applied world. They appear quite
naturally in the systems of equations I have worked with.

Furthermore, some would argue that mathematics has lost its way as it becomes
dedicated to abstraction alone.

~~~
hidenotslide
I don't understand, can you give an example?

For most of the classical applications determinants are computationally
terrible compared to factorization methods, e.g. for matrix inverse
elimination is O(n^3) and Cramer's rule is something like O(n!).

~~~
tgb
I think it's false to equate determinants with "determinants computed by
cofactor expansion". One can compute determinants efficiently through Gauss
elimination, too.

~~~
hidenotslide
Fair, but I'm still not aware of any practical applications for "systems of
equations" like the person I responded to mentioned. If you know any please
share.

The determinant intuition for me is the signed volume factor for a change of
basis. I've seen the combinatorial lattice path application and I'm sure there
are more in other fields.

But not much reason I can see to have them feature so prominently in an intro
linear algebra class. Better to spend more time with SVD for instance, which
wasn't even covered in the first linear algebra class I took.

------
adamnemecek
Obligatory mention that geometric algebra, an alternative to linear algebra,
doesn't need determinants while maintaining all the power.

------
xchip
Physicist here, don't worry if you don't understand that PDF, it is a pretty
terrible explanation.

~~~
benrbray
Why?

~~~
CamperBob2
For one thing, it uses extensive mathematical jargon that won’t make any sense
to beginners... or even to advanced students other than math majors.

~~~
wolfgke
> For one thing, it uses extensive mathematical jargon that won’t make any
> sense to beginners

In Germany this is the usual style for lectures for absolute beginners from
1st semester on - even commonly for people who don't major in math. This style
is even not uncommon for 1st semester math lectures for student who don't
major in mathematics or physics.

Hardly any faculty has a problem with this - they love it that the math
departments weed out "unsuitable" students in their lectures so they don't
have to

If you don't believe me and know a little German, here are two common German
textbooks about linear algebra covering about 1.5 semesters of linear algebra
for math majors:

\- Gerd Fischer - Lineare Algebra: Eine Einführung für Studienanfänger (note
the title "Linear Algebra: An introduction for freshmen" \- I am really not
kidding)

\- Siegfried Bosch - Lineare Algebra

Even more: I know a lecturer from Hungary who had very direct words about how
relaxing he considers the curriculum for math majors in Germany (he is used to
a Sowjet-Russian-style-inspired math program).

------
enriquto
I never understood this kind of racism against determinants.

They are very useful and intuitive, especially in 2D and 3D, where they
represent areas and volumes. For example, they give an intuitive meaning to
the notion of linear independence of 3 spatial vectors: they are independent
when they span a non-zero volume.

