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Math Help - Linear Transformations

  1. #1
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    Smile Linear Transformations

    Plese, help me to solve this:
    Let T is in L(V). Put R=Im(T) and N=null(T). Note that both R and T are T-invariant. Show that R has a complementary T-invariant subspace W (i.e. V=R (direct sum) W and T(W) is included in W) if and only if R intersect N = {0}, in which case N is the unique T-invariant subspace complementary to R.
    Thank you in advance!
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  2. #2
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    Opalg's Avatar
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    First, it follows from the rank-nullity theorem that any complementary subspace for R must have the same dimension as N.

    So if R\cap N = {0} then it's obvious that N is a T-invariant complementary subspace for R.

    Conversely, suppose that S is a T-invariant complementary subspace for R. If x\in S then Tx\in R (obviously, since Tx is in the range of T) and Tx\in S (because S is T-invariant). Therefore Tx\in R\cap S=\{0\}. In other words, x\in N. Hence S\subseteq N. But these subspaces have the same dimension, and so S = N.
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