Linear maps and composites HELP

[stukbv]

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!

 

[lanedance]

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

 

[dmatador]

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.