Uni Stuff > Mathematics
Abstract Algebra
Ahmad:
Hint:
/0:
Thanks!,
But
/0:
How would one go about proving block multiplication for general matrices? It just seems to be such a massive task, to prove it for every shape of matrix and given the complexity of the matrix multiplication formula, but it's an exercise in the book. :|
Does it require some kind of induction on matrices, how would this be done?
--- Quote from: humph on July 08, 2010, 03:45:10 pm ---Borger is a pretty good lecturer. Weird that he's changing texts though - the set text used to be Fraleigh. Dummit and Foote is supposed to be a decent abstract algebra text too.
--- End quote ---
Cool... I hope Borger doesn't endlessly copy proofs from the book like someone I won't mention...
I just discovered that Michael Artin is the son of Emil Artin, who was a famous number theorist. He also lectures at MIT, which has a reputation for good books xD
humph:
--- Quote from: /0 on July 14, 2010, 11:12:30 pm ---How would one go about proving block multiplication for general matrices? It just seems to be such a massive task, to prove it for every shape of matrix and given the complexity of the matrix multiplication formula, but it's an exercise in the book. :|
Does it require some kind of induction on matrices, how would this be done?
--- End quote ---
Induction certainly isn't the way to go. You know that you can actually write matrix multiplication in terms of summations for each entry, but it's quite ugly. I reckon the way to go would be to think of matrix multiplication in terms of the dot product, or the product of a matrix with a vector (as this gives you a scalar and a vector respectively, as opposed to a matrix), and decompose it in that way. But that might be a bit too obscure a hint.
--- Quote from: /0 on July 14, 2010, 11:12:30 pm ---
--- Quote from: humph on July 08, 2010, 03:45:10 pm ---Borger is a pretty good lecturer. Weird that he's changing texts though - the set text used to be Fraleigh. Dummit and Foote is supposed to be a decent abstract algebra text too.
--- End quote ---
Cool... I hope Borger doesn't endlessly copy proofs from the book like someone I won't mention...
I just discovered that Michael Artin is the son of Emil Artin, who was a famous number theorist. He also lectures at MIT, which has a reputation for good books xD
--- End quote ---
Oh, who's the one who copies proofs from the book?
Yeah I've gotten the two Artins confused a couple of times. Both are pretty well-reknowned for their work in algebra (and Galois theory in particular).
/0:
Thanks humph, I might return to that problem a bit later though...
I thought my Analysis professor did a lot of proof copying... but then, perhaps I'm just not used to that style of teaching. Analysis proofs give me headaches.
Anyway...
With questions like:
"Prove that the inverse of an element in a subgroup is the same as the inverse of that element in "
Are the proofs meant to be as trivial as this:
Suppose . Let and .
Then since , too.
So you have ... and by taking inverses
"Show by example that the product of elements of finite order in a nonabelian group need not have finite order."
Hmmm... I thought for a while permutations groups might work... but now I'm sure they won't, since for permutations and , there exists a permutation , which has finite order.
Pretty much the only other non-abelian group I know is ... probably not worth it?
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