Linear maps and composites HELP

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The discussion revolves around understanding the properties of a linear map T: V → V, particularly when the rank of T equals the rank of T². It emphasizes that the rank can be interpreted as the dimension of the image space of T. The main task is to demonstrate that the restriction of T to its image, denoted as Ui, is nonsingular for all i ≥ 1. The participants suggest focusing on the implications of the rank condition to establish the invertibility of Ui. Overall, the key is to connect rank properties with the non-singularity of the restricted linear maps.
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I am told that T:V--> V is a linear map where V is a finite dimensional vector space.
i also know that Ti+1 = TTi for all i >= 1 Suppose rank(T) = rank(T2)

for i>= 1 Let Ui : Im(T)-->Im(T) be defined as the restriction of Ti to the subspace Im(T) of V. Show Ui is nonsingular for all i

I have no idea what this question is asking or how to attempt it!
 
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consider the statement "rank(T) = rank(T^2)"

the rank of a matrix is equivalently:
- the number linearly independent row vectors
- the number linearly independent column vectors
- the dimension of the image space of T

consider it in terms of the third definition, meaning the image of T is the same dimension as the image of T^2
 
I think they may be asking you to show that Ui is invertible (non-singular), with some given parameters. Try to relate the statements about the rank of the matrix to properties that imply non-singularity.
 

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