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Math Help - space vector

  1. #1
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    space vector

    Check whether the data sets together with the operations indicated is a space vector. If not, list the properties that fail

    1) V={(x,y,z)/x,y,z E R} , (x1,y1,z1)+(x2,y2.z2)=(x2,y1+y2,z2) and k(x,y,z)=(kx,ky,kz)

    My Solution:

    a) u+v=v+u
    (x1,y1,z1)+(x2,y2,z2) = (x2,y1+y2,z2) = y1 + (x2,y2,z2)
    Not Check

    b) u+(v+w) = (u+v)+w
    ...
    = (x3,y1+y2+y3,z3) Not Check

    c) u+e=u
    I could not resolve

    d) u+(u')=e
    I could not resolve

    e) \alpha \beta * u= \alpha * (\beta * u)
    ...
    = \alpha * [ \beta * (x1,y1,z1)] Check

    f) (\alpha + \beta ) * u = \alpha * u + \beta * u
    ...
    = (\beta x1, \alpha y1 + \beta y1, \beta z1) Not Check

    g) 1 * u = u
    ...
    = (x1,y1,z1) Check

    h) \alpha * (u + v) = \alpha * u + \alpha * v
    ...
    = (\alpha x2, \alpha y1 + \alpha y2, \alpha z2) Not Check



    2) V={(x,y)/x,y E R} , (x1,y1)+(x2,y2)=(x1.x2,y1.y2) and k(x,y)=(kx,ky)

    a) Check
    b) Check
    c) Check
    d) I could not resolve
    e, f , g and h ) Not yet done


    Thanks
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  2. #2
    MHF Contributor arbolis's Avatar
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    There is no definition of the product of two elements in V?!
    You put a product with a scalar and an element of V, but not between 2 elements of V. Are you sure no information about the problem is missing?
    Anyway, I didn't check all but for the 1) (x1,y1,z1)+(x2,y2.z2)\neq (x2,y2,z2)+(x1,y1,z1) so V is not a vector space.
    (Detail: (x1,y1,z1)+(x2,y2,z2)=(x2,y1+y2,z2) while (x2,y2,z2)+(x1,y1,z1)=x1,y2+y1,z1).

    And about the 2), it seems to be a vector space but as you didn't define the product of two elements in V, we cannot affirm it.
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