+312 Let F be a field. A matrix = Mnxn(F)
Is tr(A)=trace(A) a member of the feild F? (Does it have to be a member of the feild?)
Just to be sure you got it, ill explain it in another way:
When you calculate tr(A) (when summing the numbers) do you refer to the members ai,i As normal numbers with the normal "+" function we all know, or add them like you add members of the feild (if in the feild 0=2 then 2+2=2)
Please answer, this is very important
+33661 Did I fail you last time?!!!
I haven't checked, and it's a long time since I did this sort of thing, but:
1. I think it wouldn't make sense to use anything other than the field's own definition of addition in calculating the trace (sum of diagonal elements).
2. Hence the trace must be a member of the field.
If the answer is important then you should double check this with someone more qualified to answer!
+312 Then ive misunderstood my professor.
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he said he can prove that if F is a field, Char(F)=0, A matrix =Â Mnxm(F), BÂ matrix =Â Mmxn(F),
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AB=In
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And
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BA=Im
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Then A is  a square matrix.
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his way: at the beggining he proved that if A matrix = Mnxm(F), B matrix = Mmxn(F) then tr(AB)=tr(BA).
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then he said n=tr(In)=tr(AB)=tr(BA)=tr(Im)=m then n=m
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BUT HE DIDNT EVEN USE THE FACT CHAR(F)=0 AND HE SAID ITS HARD TO PROVE IT WITHOUT HAVING THE FACT CHAR(F)=0. I KNOW I MISUNDERSTOOD HIM BUT MY QUESTION IS WHERE. ITS FREAKING ME OUT.
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by the way, what are you doing nowadays? are you a professor or something?
+312 Then ive misunderstood my professor.
he can prove that if F is a field, Char(F)=0, A matrix = Â Mnxm(F), B matrix = Â Mmxn(F),
AB=In
And
BA=Im
Then A is  a square matrix.
his way: at the beggining he proved that if A matrix = Mnxm(F), B matrix = Mmxn(F) then tr(AB)=tr(BA).
then he said n=tr(In)=tr(AB)=tr(BA)=tr(Im)=m then n=m
BUT HE DIDNT EVEN USE THE FACT CHAR(F)=0 AND HE SAID ITS HARD TO PROVE IT WITHOUT HAVING THE FACT CHAR(F)=0. I KNOW I MISUNDERSTOOD HIM BUT MY QUESTION IS WHERE. ITS FREAKING ME OUT.
by the way, what are you doing nowadays? are you a professor or something?
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Â
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sorry i resent it because something fucked up for some reason
+33661 I'm afraid my knowledge is insufficient here. Perhaps this will help:
http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/tracenorm.pdf
I am not and never have been a professor!
+312 Then how do you know so much about things and stuff?
What university did you graduate from?
