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/Jacobian_matrix_and_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?

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

 

[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!