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Math Help - Nulls and Ranges

  1. #1
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    Nulls and Ranges

    I got a question here that states:

    Suppose that p belongs to L(V) and P = P, show that V = nullP is a direct sum of rangeP.

    Could anyone give me a rough idea on how to approach this question? Thanks for any help.
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  2. #2
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    Quote Originally Posted by GreenDay14 View Post
    I got a question here that states:

    Suppose that p belongs to L(V) and P = P, show that V = nullP is a direct sum of rangeP.

    Could anyone give me a rough idea on how to approach this question? Thanks for any help.
    For any x in V, x = Px + (xPx). Notice that Px is in ran(P), and xPx is in null(P).

    To complete the proof, you also have to show that \text{ran}(P)\cap\text{null}(P) = \{0\}. So suppose that y = Px and that Py = 0. Can you deduce that y = 0?
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