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Math Help - Prove: If two transformations create the same map, they are equal

  1. #1
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    Prove: If two transformations create the same map, they are equal

    Let
    V be an F-vector space with basis B = (v1, v2, . . . , vn). Let W be any
    F-vector space and let w1,w2, . . . ,wn be some elements.


    Let


    S, T : V W be two linear transformations. Suppose S(vi) = T(vi) for i =1, 2, . . . , n. Show that S = T.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by amm345 View Post
    Let
    V be an F-vector space with basis B = (v1, v2, . . . , vn). Let W be any
    F-vector space and let w1,w2, . . . ,wn be some elements.


    Let


    S, T : V W be two linear transformations. Suppose S(vi) = T(vi) for i =1, 2, . . . , n. Show that S = T.
    HINT: Use linearity of mappings to show that the image of your basis forms a spanning set for the whole image.
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