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April 23rd, 2013, 08:30 PM   #1
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Finding basis for an operator of particular form

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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.
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April 24th, 2013, 07:07 PM   #2
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Re: Finding basis for an operator of particular form

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April 25th, 2013, 07:08 AM   #3
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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.
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April 25th, 2013, 10:54 AM   #4
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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.
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April 25th, 2013, 12:15 PM   #5
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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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