My Math Forum Finding basis for an operator of particular form

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 April 23rd, 2013, 08:30 PM #1 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Finding basis for an operator of particular form http://imageshack.us/photo/my-images/694/piczz.jpg/ Sorry I am unable to write the problem in the text box. It has some symbols. Please find the attached link. Can anyone help me out? Thanks a lot in advance.
 April 24th, 2013, 07:07 PM #2 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Finding basis for an operator of particular form No help yet?
 April 25th, 2013, 07:08 AM #3 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Finding basis for an operator of particular form A lot of people will not open files posted by people they do not know. There certainly isn't any reason why you could not have typed the problem in- even if you don't use latex. Let T be a nonzero operator on C^2 such that T^2= 0. Show that there exist a basis for C^2 in which the matrix for T is [ 0 1] [0 0] Since T is non-zero there exist at least one vector, v, such that Tv= u is not 0. Since T^2= 0 , Tu= T^2v= 0. Construct the matrix representing T using those two vectors as a basis for C^2.
 April 25th, 2013, 10:54 AM #4 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Finding basis for an operator of particular form Thanks HallsofIvy, thanks a lot. I sould have realized I could type that way. According to your explanation. Tv=u where u is not 0 and Tu=0. Looking at the matrix [0 1] we can say Tu=(0,0) and Tv=(1,0) [0 0] Thus u=(1,0). Since, Tv=u, we are left with finding v which makes Tv=(1,0) May I ask what would be v in case if I am correct. Thanks a lot again HallsofIvy.
 April 25th, 2013, 12:15 PM #5 Newbie   Joined: Apr 2013 Posts: 11 Thanks: 0 Re: Finding basis for an operator of particular form Sorry, HallsofIvy I guess I was wrong in the concept. Disregard the earlier post please. I have Tu=0 Tv=u Let (u,v) be the basis of C^2. In that case the matrix would have the given form since Tu=0.u+0.v and Tv=1.u+0.v So the non-zero basis would be (u,v) s.t Tu=0 and Tv=u. Can you define any such T? Thanks again.

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