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Math Help - Isomorphism and dimensional vector spaces

  1. #1
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    Isomorphism and dimensional vector spaces

    1. Let U,V be finite dimensional vector spaces,with bases B,C respectively.
    Prove:a linear map f:U->V is an isomorphism iff [f]_C^B is invertible.(I'm so sorry but I honestly couldn't get Latex to work properly,slightest change to "[f]" and it would cause an unknown latex error)

    2. Let T:M_n(R)->T:M_n(R),T(A)=AC,C \in M_n(R).
    Prove that it is an isomorphism iff C is invertible.

    Again so sorry for the inconvenience and any help would be appreciated.
    Thanks in advance
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  2. #2
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    I presume that [f]_C^B means the matrix representation of f using those bases.

    An isomorphism is, by definition, both one to one and onto. If the vector y is in V then there exist a unique x in U such that f(x)= y. That says that, for any y in v, there exist a unique x such that [f]_C^Bx= y. It follows immediately that kernel of f is just {0} and from that that the matrix is invertible.
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  3. #3
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    for problem 2, suppose C is invertible, and let A be any 2x2 real matrix. then T(AC^-1) = (AC^-1)C = A(C^-1C) = AI = A, so

    A has the pre-image AC^-1, thus T is onto.

    suppose T(K) = KC = 0. then (KC)C^-1 = 0C^-1 = 0. but (KC)C^-1 = K, whence K = 0, so T is 1-1. thus T is an isomorphism

    (the proof that T is R-linear is rather straight-forward).

    on the other hand, suppose T is an isomorphism. since all isomorphisms are invertible maps, T^-1 exists. since T is onto,

    I is in the domain of T^-1. let B = T^-1(I).

    then BC = T(B) = T(T^-1(I)) = I, so B = C^-1, C is invertible.
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