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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

Ehrlich  Mar 25, 2017
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7+0 Answers

 #1
avatar+25984 
+2

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!

Alan  Mar 25, 2017
 #2
avatar+310 
-1

Then ive misunderstood my professor.

 

he said he can prove that if F is a field, Char(F)=0, A matrix =  Mnxm(F), B matrix =  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?

Ehrlich  Mar 25, 2017
 #3
avatar+310 
-1

Then ive misunderstood my professor.

he can prove that if F is a field, Char(F)=0, A matrix =  Mnxm(F), B matrix =  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?

 

 

 

sorry i resent it because something fucked up for some reason

Ehrlich  Mar 25, 2017
 #4
avatar+25984 
0

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!

Alan  Mar 25, 2017
 #5
avatar+310 
-1

Then how do you know so much about things and stuff?

 

What university did you graduate from?

Ehrlich  Mar 25, 2017
 #6
avatar+25984 
+2

"Then how do you know so much about things and stuff?"  

I don't know very much about anything!  I've applied math to real world systems for a long time though.

 

"What university did you graduate from?"

Warwick University  (in the UK)

Alan  Mar 25, 2017
 #7
avatar+310 
+1

Atleast there is a reliable source in this site then :D

Ehrlich  Mar 25, 2017

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