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Math Help - Help with elementary Linear Algebra proof?

  1. #1
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    Help with elementary Linear Algebra proof?

    I've been brushing up on some linear algebra this summer, and here's a problem I found in a book that I can't seem to solve. Any help is appreciated!

    Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If the intersection of R(T) and R(U)* is 0, prove that {T, U} is a linearly independent subset of L(V, W).**

    *R(T) and R(U) denote the range of the linear transformation of T and U.
    ** L(V, W) denotes the space of linear transformations from V to W.
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  2. #2
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    Quote Originally Posted by paupsers View Post
    I've been brushing up on some linear algebra this summer, and here's a problem I found in a book that I can't seem to solve. Any help is appreciated!

    Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. If the intersection of R(T) and R(U)* is 0, prove that {T, U} is a linearly independent subset of L(V, W).**

    *R(T) and R(U) denote the range of the linear transformation of T and U.
    ** L(V, W) denotes the space of linear transformations from V to W.
    suppose they are not linearly independent. then T=cU, for some scalar c. since T \neq 0, there exists v \in V such that T(v) \neq 0. but we also have T(v)=cU(v)=U(cv), which gives us:

    T(v) \in R(T) \cap R(U) = (0). contradiction!
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