The 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 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.
Another 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.
At 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).
Seriously, 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?
I 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.
Understanding 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.
Yes. 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.
- 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.