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Math Help - help me on invariant subspace please

  1. #1
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    help me on invariant subspace please

    let T denote a linear operator on a vector space V. suppose that every subspace of V is invariant under T. prove that T is a scalar multiple of the identity map.

    please help me.
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    let T denote a linear operator on a vector space V. suppose that every subspace of V is invariant under T. prove that T is a scalar multiple of the identity map.
    Step 1: Let x be a nonzero vector in V. Then the one-dimensional subspace spanned by x is invariant under T. So Tx = c_xx, for some scalar that (possibly) depends on x.

    Step 2: Let x and y be linearly independent vectors in V. Then Tx = c_xx,\ Ty = c_yy, \ T(x+y) = c_{x+y}(x+y). Deduce from this that c_x = c_y. Thus the constant c is in fact the same for every vector, and hence T is c times the identity.
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  3. #3
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    thanks

    thank you very much. you really helped me.
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