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Math Help - Question involving Vector Spaces

  1. #1
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    Question involving Vector Spaces

    Let V be the set of all ordered pairs of real numbers with addition defined by

    (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2)

    and scalar multiplication defined by

    α(x1, x2) = (αx1, x2)

    The scalar multiplication for this system is defined in an unusual way, and consequently we use the symbol to avoid confusion with the ordinary scalar multiplication of vectors. Is V a vector space with these operations? Justify your answer.


    I honestly would think not because α(x1, x2) should equal (αx1, αx2) however the problem states it is not ordinary scalar multiplication. It doesn't state what kind though . Help would be appreciated, thank you!
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    What is meant by "The scalar multiplication for this system is defined in an unusual way" is that you do not use \alpha(x_1,x_2)=(\alpha x_1,\alpha x_2) (which is the classical product) but \alpha(x_1,x_2)=(\alpha x_1,x_2), that's all. The product \alpha x_1 is the usual product of real numbers.
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  3. #3
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    earboth's Avatar
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    Quote Originally Posted by pakman View Post
    Let V be the set of all ordered pairs of real numbers with addition defined by

    (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2)

    and scalar multiplication defined by

    α(x1, x2) = (αx1, x2)

    The scalar multiplication for this system is defined in an unusual way, and consequently we use the symbol to avoid confusion with the ordinary scalar multiplication of vectors. Is V a vector space with these operations? Justify your answer.


    ...
    I assume that you know the axioms of a vector space.

    You only have to check if all (8) axioms are satisfied by the given definitions.
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