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September 24th, 2012, 10:01 AM  #1 
Newbie Joined: Sep 2012 Posts: 1 Thanks: 0  Abstract Algebra Help
1) Find two linear maps A,B: R2 > R2 such that A°B ? B°A I understand how to find the two linear maps, but I am still lost with respect to "such that A°B ? B°A" 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. 
September 24th, 2012, 11:07 PM  #2 
Senior Member Joined: Nov 2011 Posts: 595 Thanks: 16  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 ) A°B(xo) is different than B°A(xo) Then for the second start with a(u+v)+b(uv)=0. The aim is to show that a=b=0 then. You get (a+b)u+(ab)v=0. Since u and v are independent this leads to a+b=0 and ab=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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