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Math Help - linear algebra

  1. #1
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    linear algebra

    If T:V-->V is normal.

    Prove a.) null ( T^k) = null (T)


    b.) range( T^k)=range(T)
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  2. #2
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    i know i need to prove that if

    x \in null (T)

    therefore, T(x)=0

    but i'm unsure as to how i get towards the other side. I know the operator is normal so it is diagonalizable in some basis due to the spectral theorem.

    ......?
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  3. #3
    Junior Member
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    ok so i settled on this as my way to solve the first part.

    Since, T is normal, there exists an ONB of V so that the matrix representation of T is diagonal (by the complex spectral theorem)

    Since A is diagonal, the null(A)=0 (the zero vector).

    T^k can be represented as A but with its entries to the power of k.

    Since T^k can be represented as A' which is also diagonal, the null( T^k)=0 the zero vector.

    Therefore the null(T)=null( T^k)=0 the zero vector.

    ****AM I ON THE RIGHT TRACK?!?!?!!*******

    also, how do i prove the range side? thanks all
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