http://www.axler.net/DwD.html

#1 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.
I suffered through determinants.

#2 omazurov:
Determinants have a very basic intuition behind them: it's the stretch factor of the nvolume of a linearly transformed unit ncube (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?

#3 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...
It's mostly graphical, and is really helpful in forming and cementing an intuition for linear algebra.

#4 experiment0:
This is a great paper.
As a counterpoint, one place where determinants are incredibly useful is in HartreeFock theory, where they effective encode the Pauli exclusion principle (or antisymmetry requirements) of atomic orbitals.

#5 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!

#6 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.

#7 pacala:
> A complex number λ is called an eigenvalue of T if T −λI is not injective.
Uhm, what the intuition behind _that_?

#8 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.

#9 undefined:
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#10 adamnemecek:
Obligatory mention that geometric algebra, an alternative to linear algebra, doesn't need determinants while maintaining all the power.

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

#12 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 nonzero volume.