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Math Help - Isomorphism

  1. #1
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    Isomorphism

    Let V and W be linear spaces onto field F.

    T:V \rightarrowW is called Isomorphism from V over W iff it has these three properties:
    1) T is a linear transformation
    2) T is one-to-one
    3) T is onto W

    So by that definition of Isomorphism can I say that if KerT has more than one vector that transforms to 0_w then those subspaces are not isomorphic.

    Basically, I'm concluding that if two spaces are isomorphic to each other, only the zero vector of V transforms to the zero vector of W
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by jayshizwiz View Post
    Let V and W be linear spaces onto field F.

    T:V \rightarrowW is called Isomorphism from V over W iff it has these three properties:
    1) T is a linear transformation
    2) T is one-to-one
    3) T is onto W

    So by that definition of Isomorphism can I say that if KerT has more than one vector that transforms to 0_w then those subspaces are not isomorphic.

    Basically, I'm concluding that if two spaces are isomorphic to each other, only the zero vector of V transforms to the zero vector of W
    u are right since if the kernel have more than two vectors say u,v u\ne v \; then
    T(u) = T(v) = 0 not one-one
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