document.write( "Question 1136203: suppose that V is finite-dimensional space and T:V->V is a linear operator.
\n" ); document.write( "1. prove that rank(T^k+1) <= rank(T^k)
\n" ); document.write( "2. Prove that there exists k in N such that rank(T^k+1)=rank(T^k)
\n" ); document.write( " I know that R(T^k+1) ⊆ R(T^k) for any k. We can choose v ∈ R(T^k+1).Then for some c, (T^k+1)c = v. But T^k(Tc) = v, so v ∈ R(T^k). How would I proceed with this? any help is appreciated. \n" ); document.write( "

Algebra.Com's Answer #753972 by ikleyn(52803) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " Unfortunately, I don't know, what exactly is your level in Linear Algebra.\r
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document.write( "You have a linear map (operator) T from a finite-dimensional linear space V to itself.\r\n" );
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document.write( "rank(T)  is the dimension of the image of the space V under this transformation.\r\n" );
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document.write( " are the degrees of the operator T, what you can interpret as sequential iterations of T.\r\n" );
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document.write( "Then it is clear that  can not rise up.  \r\n" );
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document.write( "It only can go down - not necessary strictly down at each step/iteration.\r\n" );
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document.write( "Not necessary in monotonic way down. But not rise up, in any case.\r\n" );
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document.write( "There are two typical examples.\r\n" );
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document.write( "One example is when the matrix of T is zero everywhere, except one diagonal above the major diagonal.\r\n" );
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document.write( "    In this case,    decreases monotonically at each step/iteration till 0 (zero) \r\n" );
document.write( "    after n iterations, where n = dim(V).\r\n" );
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document.write( "The other example is an operator of projection to a subspace.  Than  stabilizes on some positive value at some step.\r\n" );
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document.write( "    (and this value is, OBVIOUSLY, the dimension of the image of T).\r\n" );
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document.write( "So, after my explanations, n.1 is just proved/explained (using my fingers).\r\n" );
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document.write( "n.2 also becomes evident now, meaning stabilization of values of .\r\n" );
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document.write( "    This stabilization can be achieved at some positive value of , or at the zero value \r\n" );
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document.write( "    (which means then that the operator T and its matrix is/are nilpotent).\r\n" );
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