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Math Help - Linear Operator question

  1. #1
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    Linear Operator question

    Hello. I need confirmation on this: Let T:V\rightarrow V be a linear operator on the (finite dimensional)vector sapce V. I know that when T is a proyector,ie, T^2=T, then the space is the direc sum of the Kernel of T and it's image, V=ker(T)\oplus Im(T). Now it seems to me that this is also true for any operator( T^2\neq T). am I correct?
    Last edited by facenian; August 20th 2010 at 10:54 AM. Reason: correction
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    No! For example, what if you have T^2=0 but T \neq 0? It's true that the dimensions always add up, but the direct sum decomposition is a much stronger statement.
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  3. #3
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    Thank you. With your help I think I found the mistake in my proof.
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