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September 24th, 2012, 10:01 AM   #1
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Abstract Algebra Help

1) Find two linear maps A,B: R2 ---> R2 such that AB ? BA

I understand how to find the two linear maps, but I am still lost with respect to "such that AB ? BA"

2) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Z2.

For this one, I am just completely confused. I understand the idea of proving they are linearly independant but I am having trouble prooving if the same conclusion would be true over Z2.

Any help would be greatly appreciated.
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September 24th, 2012, 11:07 PM   #2
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Re: Abstract Algebra Help

For the first one, use for instance:
A: Rotation by 90 so (x,y)->(-y,x)
B: Translation by a vector u=(1,0) so (x,y)->(x+1,y)
You can check for instance taking an initial point xo=(1,0 ) AB(xo) is different than BA(xo)

Then for the second start with a(u+v)+b(u-v)=0. The aim is to show that a=b=0 then. You get (a+b)u+(a-b)v=0. Since u and v are independent this leads to a+b=0 and a-b=0. So a=b=0. I did also not understand what is the different if it is Z2 or R2 so probably check this again, perhaps I missed something there (or not)..
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