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Ooh I'd be excited to read this. I recently found "Linear Algebra Done Right." How, you ask? They defer using determinants until the end of the book. Determinan
by delish 11y ago
Ooh I'd be excited to read this. I recently found "Linear Algebra Done Right." How, you ask? They defer using determinants until the end of the book. Determinants are hard computationally, hard to define, and not obviously useful. Without them (and I haven't read the book but I will), one has to make more general, less computational proofs.
- stsp 11y ago"Linear Algebra Done Right" helped me with getting through college :) I can highly recommend it.
- jules 11y agoThe determinant of a linear map is its volume scale factor: any shape of volume V in the input space gets transformed to a shape of volume det(L)V in the output space. This makes a bunch of things easy to understand: - Why you can calculate the volume of a slanted box with a determinant - Why a matrix is singular if its determinant is zero - Why the determinant is the product of the eigenvalues - Why det(I) = 1, why det(AB) = det(A)det(B), and why det(A^-1) = 1/det(A) - Why change of variables in integration gets a det(J) factor Sadly, most linear algbra texts introduce the determinant as some random summation formula or with a series of unmotivated axioms. This is a general problem with mathematics: symbols over geometry, and formal proofs over intuitive understanding when it should have been the other way around.
- torrance 11y agoSeriously, thank you for this. This makes so much of what I've been learning make sense and helps with the intuition around determinants. Why isn't this the first thing they teach about determinants?
- kuschku 11y agoI asked a friend, and he said "but isn’t it obvious that if you sum up this and this and...". Yes. To many mathematicians, it is just obvious. Maybe that’s the issue.
- technomancy 11y agoUnderstanding what is and what isn't obvious to other people is a good chunk of what makes the difference between a good teacher and a bad one--possibly the single most important factor. Yet it has nothing to do with what makes a good mathematician.
- kuschku 11y agoYes. But obviously, at a university, most professors are selected for their research. Which does not, unless they research teaching, necessarily mean they are good at teaching.
- a_e_k 11y agoAt least in 3-space, they also have an interesting relationship with signed tetrahedral volumes, the scalar triple product, and Pluecker coordinates. I once got a published paper out of those connections (http://www.cs.utah.edu/~aek/research/triangle.pdf http://www.cs.utah.edu/~aek/research/triangle.pdf).
- jules 11y agoNice, and that generalizes neatly to n-space with exterior algebra / Grassmann coordinates.
- speaknotyet 11y agoAnother application of determinants: - Compute a vector perpendicular to other two vectors. - Compute the rotational of a vector field - Determine if three vector are linearly dependent - If you consider a complex number z = a + bi, then there is a linear transformation from C to C given by x -> xz, than you can see as a linear transformation from R^2 to R^2, the determinant of that matrix is the amplifying factor for the area of figures.