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Math Help - Basis of a Cartesian product

  1. #1
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    Basis of a Cartesian product

    Let U, V be finite dimensional linear spaces and let \{u_1,...,u_n\} be a basis for U and let \{v_1,...,v_m\} be a basis for V. Let W=U\times V. Construct a basis for W.

    I'm probably making this way harder than it really is. The only idea I came up with was

    \{(u_i, v_j) | 1\leq i\leq n, 1\leq j\leq m\}

    But I know that's wrong....
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by mathematicalbagpiper View Post
    Let U, V be finite dimensional linear spaces and let \{u_1,...,u_n\} be a basis for U and let \{v_1,...,v_m\} be a basis for V. Let W=U\times V. Construct a basis for W.

    I'm probably making this way harder than it really is. The only idea I came up with was

    \{(u_i, v_j) | 1\leq i\leq n, 1\leq j\leq m\}

    But I know that's wrong....
    Think of an example. Take your favourite 2-dimensional vector space and cross it with your favourite 3-dimensional vector space. Can you think of a basis for this new space?

    Also, I would recommend contemplating how many elements your new basis will have. Hopefully the example will make this clear, but if not...think about it...
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Isomorphism View Post
    To solve your problem, consider a set with the elements of the form -Blank-.
    Don't just give the answer away! Rule 14!
    Last edited by Swlabr; August 26th 2010 at 01:20 AM.
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