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Math Help - Isomorphic Vector Spaces

  1. #1
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    Isomorphic Vector Spaces

    Here is a question, I have came up with an awnser, although im not really sure if my proof is sufficient.

    Let M_{\infty}(R) denote the vector space of infinite matrices with finitely many non zero enteries.

    Let R^{\infty} be the vector space of infinite sequences in which only finitely many elements are nonzero.

    Prove that R^{\infty} and M_{\infty}(R) are isomorphic.

    My attempt at the solution:

    We know two vector spaces are isomorphic, if and only if their dimensions are equal.

    Let m = the finite number of non zero entires in M_{\infty}(R). This is equal to the dimension of M_{\infty}(R).

    Let r = the finite number of non zero elements in R^{\infty}. This is equal to the dimension of R^{\infty}

    Since r and m are just any finite number, we can let r = m, and then since the dimensions are equal, they are isomorphic.

    Is this proof enough? If not a point in the right direction would be great! Thanks.
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  2. #2
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    Quote Originally Posted by joe909 View Post
    Here is a question, I have came up with an awnser, although im not really sure if my proof is sufficient.

    Let M_{\infty}(R) denote the vector space of infinite matrices with finitely many non zero enteries.

    Let R^{\infty} be the vector space of infinite sequences in which only finitely many elements are nonzero.

    Prove that R^{\infty} and M_{\infty}(R) are isomorphic.

    My attempt at the solution:

    We know two vector spaces are isomorphic, if and only if their dimensions are equal.

    Let m = the finite number of non zero entires in M_{\infty}(R). This is equal to the dimension of M_{\infty}(R).

    Let r = the finite number of non zero elements in R^{\infty}. This is equal to the dimension of R^{\infty}

    Since r and m are just any finite number, we can let r = m, and then since the dimensions are equal, they are isomorphic.

    Is this proof enough? If not a point in the right direction would be great! Thanks.

    There is no fixed number of non-zero entries either in M_\infty(\mathbb{R}) \,\,\,or\,\,\,in\,\,\,\mathbb{R}^\infty ! So your numbers m, r are meaningless.

    In fact, both space are infinite dimensional, but the gist of the matter is that both have the same cardinal as dimension, which is = \aleph_0 over the reals \mathbb{R} and, thus, they both are isomorphic.

    Tonio
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  3. #3
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    Quote Originally Posted by joe909 View Post
    Here is a question, I have came up with an awnser, although im not really sure if my proof is sufficient.

    Let M_{\infty}(R) denote the vector space of infinite matrices with finitely many non zero enteries.

    Let R^{\infty} be the vector space of infinite sequences in which only finitely many elements are nonzero.

    Prove that R^{\infty} and M_{\infty}(R) are isomorphic.

    My attempt at the solution:

    We know two vector spaces are isomorphic, if and only if their dimensions are equal.
    No, we do not know that! That is only true for finite dimensional vector spaces.

    Let m = the finite number of non zero entires in M_{\infty}(R). This is equal to the dimension of M_{\infty}(R).
    That doesn't even make sense. Individual matrices in M_\infty have different numbers of non zero entries. There is no one "m" for every matrix. Since there is no upper bound to how many non-zero entries such a matrix can have, this vector space is infinite dimensional.

    Let r = the finite number of non zero elements in R^{\infty}. This is equal to the dimension of R^{\infty}
    Again, that does not make sense. Different sequences have different numbers of non-zero entries. Since there is no upper bound to the number of non-zero entries a sequence can have, this vector space is infinite dimensional.

    Since r and m are just any finite number, we can let r = m, and then since the dimensions are equal, they are isomorphic.

    Is this proof enough? If not a point in the right direction would be great! Thanks.
    No, you have apparently misunderstood the what is meant by "finite number of non-zero entries. You will need to find an actual isomorphism from M_\infty to R^\infty. Think about making the entries in a matrix into a "list" by zig-zagging through the matrix.
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