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Homework Help: Vector Spaces Question

  1. Sep 27, 2005 #1

    mpm

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    I have a homework problem that I can't figure out and there is nothing in the book that helps me out. I was hoping someone could shed some light.

    Let R^+ denote the set of postive real numbers. Define the operation of scalar muplication, denoted * (dot) by,

    a*x = x^a

    for each X (episilon) R^+ and for any real number a. Define the operation of addition, denoted +, by

    x + y = x * y for all x, y (Epsilon)R^+

    Thus for this system teh scalar product of -3 times 1/2 is given by

    - 3 * 1/2 = (1/2)^-3 = 8

    and the sume of 2 and 5 is given by

    2 + 5 = 2 * 5 = 10

    Is R^+ a vector space with these operations? Prove your answer.

    The plus should be a plus with a circle around it but I couldnt figure out how to put it in there. Im also not sure how to make the epsilon either.

    Any help would be greatly appreciated.

    mpm
     
    Last edited: Sep 27, 2005
  2. jcsd
  3. Sep 27, 2005 #2

    AKG

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    Science Advisor
    Homework Helper

    This is an epsilon ([itex]\epsilon[/itex]), you're looking for a different symbol, the "is a member or element of" relation ([itex]\in[/itex]). So you have:

    [tex](\mathbb{R}^+, \oplus, \otimes)[/tex]

    with the following definitions, for all x, y in R+ and all scalars (reals) [itex]\lambda[/itex]:

    [tex]x \oplus y = x \times y[/tex]

    [tex]\lambda \otimes x = x^{\lambda}[/tex]

    Do you know the definition of a vector space? Basically, all you have to do is check that the operations are well-defined, and then show that they satisfy all the properties (like commutativity of addition, associativity of scalar multiplication, existence of identities, etc.).
     
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