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Math Help - Counter example R(T+U) subset of or equal R(T) + R(U)

  1. #1
    Newbie Nona's Avatar
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    Counter example R(T+U) subset of or equal R(T) + R(U)

    Hello everyone,
    I need help plz

    Find a Counter example for:
    If T and U are linear transformations from V to W and W are vector space then R(T+U) subset of or equal R(T) + R(U).

    Thank you
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  2. #2
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    What is R?
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  3. #3
    Newbie Nona's Avatar
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    R = Rank
    Thank you
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  4. #4
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    The rank of a transformation (or matrix) is a number. How can a number be a subset of two other numbers?

    Perhaps you wanted to find a counter example of

    rank(T+U) \leq rank(T) + rank(U) ?
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  5. #5
    Newbie Nona's Avatar
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    oops
    R(T) = the range of the linear transformation T
    sorry
    Thanks
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  6. #6
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    Quote Originally Posted by Nona View Post
    Hello everyone,
    I need help plz

    Find a Counter example for:
    If T and U are linear transformations from V to W and W are vector space then R(T+U) subset of or equal R(T) + R(U).

    Thank you
    There is no counterexample for this since it's true: y=Tx+Ux \in R(T)+R(U). The other inclusion is false: take T,U: V \rightarrow V with T=id_V and U=-id_V then R(T+U)= \{0 \} and R(T)+R(U)=V
    Last edited by Jose27; December 4th 2009 at 03:43 PM.
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