Determinants

[matqkks]

How can I make something like determinants tangible? Are there real life examples where determinants are used?

[vanesch]

How can I make something like determinants tangible? Are there real life examples where determinants are used?

Volume comes up:

- The volume of a parallelepiped equals the determinant of the 3 x 3 matrix made up by its 3 "base vectors".
(if "oriented" lefthanded, it will come out negative).

[Klockan3]

The determinant determines how the corresponding linear mapping changes all kinds of volumes.

[Monocles]

Volume is probably the most tangible example, but another one to keep in mind is that they allow you to determine whether a system of linear equations has a solution (the determinant of the matrix of coefficients must be nonzero). To apply this to real life you just need to come up with a real life situation where you get a system of linear equations. On the other hand, though, when first introducing the determinant to someone this might just seem like voodoo.

[mathwonk]

the two are related. and the explanation is a little backwards. having determinant zero is not necessary for a solution to exist but rather it is sufficient.

(this discussion only applies to maps between spaces of the same dimension.)
if the determinant is non zero, then an n dimensional block is transformed into another n dimensional block, i.e. the dimension of the image space is the same as that of the source space.

It follows that the image space is equal to the entire target space, and hence that every equation has a solution. On the other hand even if a linear map from n space to n space lowers dimension, so that the image is a proper subspace of the target, some equations will still have solutions, but not all.

[victor.raum]

How's about considering the way determinants were discovered to begin with, namely in relation to finding the solution point for a set of N N-dimensional linear equations. That's certainly tangible :-)