I strongly disagree skipping determinants provides a more intuitive approach to linear algebra. I don't know your background, but I'd venture a guess you feel it does because the Laplace expansion formula for computing the determinant[1] feels uninspired and out of place.
The reason determinants are hard to teach (in my opinion) is because a rigorous derivation of their formula isn't possible without first teaching multilinear algebra and constructing the exterior algebra. Once you do those things, the natural geometric interpretation of the determinant basically falls onto your lap. But it's still very useful for e.g. computing eigenvalues and using the characteristic polynomial, so it's taught before that context can be formalized.
Professors shouldn't teach determinants in the context of matrices, at least not at first. That's heavily computation-focused, and the symbol pushing looks really unmotivated and strange to students. Instead they should teach the basis-free definition of determinants (i.e. focus on the linear map, not the matrix transformation representing the linear map for some basis). Then the determinant is "only" the volume of the image of the unit hypercube under the linear transformation, which is where the parallelepiped comes in. If the linear transformation is invertible, the unit hypercube is transformed from an n-dimensional cube into an n-dimensional parallelogram, from which you can geometrically see the way the linear map transforms the entire vector space it's defined over.
3Blue1Brown has a very good video on the geometry underlying the determinant[2]. For a more rigorous presentation which constructs the exterior algebra and derives the determinant formula using the wedge product, Noam Elkies has notes[3][4] for when he teaches Math 55A at Harvard. Incidentally Noam Elkies uses Axler's book, and while he obviously approves of it he's pretty upfront in asserting that the determinant should be taught anyway[5].
I agree, the way I still see determinant is as the 'volume scaling factor' of a linear transformation.
This means it makes sense that det(A) = 0 means A is non-invertible. It also makes a lot of sense when the jacobian pops up in the multi-dimensional chain rule.
Given the above, and the Cayley–Hamilton theorem, I never really had to know why the determinant was calculated the way it is. The above give enough of an interface to work with it.
The reason determinants are hard to teach (in my opinion) is because a rigorous derivation of their formula isn't possible without first teaching multilinear algebra and constructing the exterior algebra. Once you do those things, the natural geometric interpretation of the determinant basically falls onto your lap. But it's still very useful for e.g. computing eigenvalues and using the characteristic polynomial, so it's taught before that context can be formalized.
Professors shouldn't teach determinants in the context of matrices, at least not at first. That's heavily computation-focused, and the symbol pushing looks really unmotivated and strange to students. Instead they should teach the basis-free definition of determinants (i.e. focus on the linear map, not the matrix transformation representing the linear map for some basis). Then the determinant is "only" the volume of the image of the unit hypercube under the linear transformation, which is where the parallelepiped comes in. If the linear transformation is invertible, the unit hypercube is transformed from an n-dimensional cube into an n-dimensional parallelogram, from which you can geometrically see the way the linear map transforms the entire vector space it's defined over.
3Blue1Brown has a very good video on the geometry underlying the determinant[2]. For a more rigorous presentation which constructs the exterior algebra and derives the determinant formula using the wedge product, Noam Elkies has notes[3][4] for when he teaches Math 55A at Harvard. Incidentally Noam Elkies uses Axler's book, and while he obviously approves of it he's pretty upfront in asserting that the determinant should be taught anyway[5].
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1. http://mathb.in/33068
2. https://www.youtube.com/watch?v=Ip3X9LOh2dk
3. http://www.math.harvard.edu/~elkies/M55a.10/p8.pdf
4. http://www.math.harvard.edu/~elkies/M55a.10/p9.pdf
5. http://www.math.harvard.edu/~elkies/M55a.10/index.html