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You use the same formula to explicitly calculate the determinant in abstract proof. How can a student understand that, for example, the volume form on a rieman

by hansen 16y ago

You use the same formula to explicitly calculate the determinant in abstract proof. How can a student understand that, for example, the volume form on a riemannian manifold is invariant under a change of the chart, w/o knowing how to explicitly calculate a determinant? We all hate cumbersome and (at a first sight) useless calculations, but it's the easiest way to become familiar with the math behind it.

tome 16y ago
You really ought not to use the formula for anything other than calculation. You should characterise the determinant as the highest order form in the Exterior Algebra[1] and then use the fact that it's a group homomorphism so det(UAU^{-1}) = det(A) That's not to say that I agree with the GP. There's a lot to be said for testing the ability to perform abstract calculations with no practial use. A dimension 6 determinant seems completely unreasonable though. [1] http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality
- ghurlman 16y ago
  It's posts like this that remind me how now, 10 years out of college in my software dev career, just how much math I've completely forgotten. I like to think I'd pick it up again quickly if need be - but that's the thing... I've forgotten it, because I've never needed it.
- yummyfajitas 16y ago
  In an intro linear algebra class, I think a 6x6 determinant is legit. I asked one on my midterm. It was actually a very easy problem since the matrix was about 2 row reductions steps away from being upper triangular. Unless the professor is evil, "compute this 6x6 determinant" is really asking "do you know the properties of determinants which will simplify this problem?"
drewcrawford 16y ago
My point was not to say that familiarity with the determinant formula was worthless. Potentially, with an infinite amount of time and scratch paper, etc., doing the determinant of one 6x6 in a homework setting might be a worthwhile activity. However, presenting a nonzero nonsparse 6x6 as a "break" between two formal proofs on a large timed exam is not my idea of the level of arithmetic agility that should be expected of an undergrad CS student. No doubt there are great insights to be gained from application of the determinant formula. One that occurred to me at the time was "Computationally, what is the best order to perform these operations to get a numerically exact result? What is the time complexity of a perfect algorithm to determine the best-exact order? Is there a faster approximation that still provides good results?" etc. Unfortunately, I never pursued any of these arguably-more-relevant questions, because I was busy getting my rote multiplication up to speed. I never even thought of some of your applications, because I was too busy with nonsense to make connections. While this is a point example, it in many ways is representative of my math education.

by hansen 16y ago

tome 16y ago

You really ought not to use the formula for anything other than calculation. You should characterise the determinant as the highest order form in the Exterior Algebra[1] and then use the fact that it's a group homomorphism so det(UAU^{-1}) = det(A) That's not to say that I agree with the GP. There's a lot to be said for testing the ability to perform abstract calculations with no practial use. A dimension 6 determinant seems completely unreasonable though. [1] http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality

ghurlman 16y ago
It's posts like this that remind me how now, 10 years out of college in my software dev career, just how much math I've completely forgotten. I like to think I'd pick it up again quickly if need be - but that's the thing... I've forgotten it, because I've never needed it.
yummyfajitas 16y ago
In an intro linear algebra class, I think a 6x6 determinant is legit. I asked one on my midterm. It was actually a very easy problem since the matrix was about 2 row reductions steps away from being upper triangular. Unless the professor is evil, "compute this 6x6 determinant" is really asking "do you know the properties of determinants which will simplify this problem?"

drewcrawford 16y ago

My point was not to say that familiarity with the determinant formula was worthless. Potentially, with an infinite amount of time and scratch paper, etc., doing the determinant of one 6x6 in a homework setting might be a worthwhile activity. However, presenting a nonzero nonsparse 6x6 as a "break" between two formal proofs on a large timed exam is not my idea of the level of arithmetic agility that should be expected of an undergrad CS student. No doubt there are great insights to be gained from application of the determinant formula. One that occurred to me at the time was "Computationally, what is the best order to perform these operations to get a numerically exact result? What is the time complexity of a perfect algorithm to determine the best-exact order? Is there a faster approximation that still provides good results?" etc. Unfortunately, I never pursued any of these arguably-more-relevant questions, because I was busy getting my rote multiplication up to speed. I never even thought of some of your applications, because I was too busy with nonsense to make connections. While this is a point example, it in many ways is representative of my math education.