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Math Help - Prove isomorphism

  1. #1
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    Prove isomorphism

    Let a_0,...,a_n be distinct real numbers.
    Let the linear map T: R[X]_{\leq n} \rightarrow R^{n+1} be defined by  T(q(x)) = [q(a_0),....,q(a_n)]^T for all q(x) in R[X]_{\leq n}

    Prove that T is an isomorphism

    For this question is it enough to say that R[X]_{\leq n} and R^{n+1} are both of dimension n+1 and therefore isomorphic? Im not sure i can apply this to a transformation. Thanks for any help!
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  2. #2
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    No. These two are isomorphic does not show that the map T is an isomorphism. However, since these two spaces have the same dimension, it suffices to prove that the kernel (or nullspace) of T is trivial, which is easy (but not so trivial) to prove.
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  3. #3
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    Well its clear that the only polynomial that gives 0 for all values of x is the 0 polynomial therefore the kernal of T is trivial. Im confused as to how knowing that the kernal of T consist of only the 0 polynomial. And since the kernal of T is 0, it is therefore invertible. Which makes it isomorphic? Is this correct?

    Thanks!
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