The importance of determinants in linear algebra.

matqkks

In some literature on linear algebra determinants play a critical role and are emphasized in the earlier chapters. (See books by Anton & Rorres, and Lay). However in other literature it is totally ignored until the latter chapters. (See Gilbert Strang).
How much importance should we give the topic of determinants . I tend to use it to find linear independence of vectors and might extend this to finding the inverse but I think Gauss Jordan and LU might be easier for inverse. Does it have any other uses in Linear Algebra.
Are there areas where determinants are used and have a real impact? Are there any real life applications of determinants?
Is there a really good motivating example or explanation which will hook students into this topic?

M Quack

See example 1

http://en.wikipedia.org/wiki/Jacobia...nd_determinant

Not really pure linear algebra, but what in real life is?

theorem4.5.9

Determinants are used all over the place, not only in linear algebra. One of the uses of determinants that comes up a lot in my studies is its use in computing areas.

In calc 3 you learned (or will learn) that in order to do a u-substitution in 3 dimensions, you need to multiply dx by the jacobian, which is a determinant. It's the infinitesimal change of area. This idea crops up all the time in certain areas of geometry.

matqkks

Sorry but on a linear algebra where should determinants be placed?
Like I sain in my comment - in some literature it is at the beginning whilst in others it is bolted on at the end. I like the idea of checkiing if vectors are independent by using determinants so think they should be placed before independence of vectors.
What do you think? If you teach a linear algebra course where do you place this topic.

M Quack

I would probably put them closer to the end. Checking linear independence is nice, but then it kind of hangs around unused for a long time. You really need them when you get to eigenvalues and the characteristic polynomial.

theorem4.5.9

Contrary to how math is typically presented, most subjects are not linear in pedagogy. Your question is about preference, and that changes with different people and authors. Personally I like to use them early because they have a very geometric description to them.

mathwonk

it depends on your focus on computations. if you want to actually compute minimal polynomials, it helps to know this theorem.

Skrew

Computing a determinant of any real size is very computationally intensive.

matqkks

Which Theorem are you referring to?

Mandlebra

The book linear algebra done right avoid the use of determinants until the very end. The proofs are done without t he determinant. If det makes you uneasy, check it out!