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Math Help - Linear Algebra space

  1. #1
    cris123
    Guest

    Linear Algebra space

    i did the work just i need more details and justify by stating axions and theorems,, please help

    2) Let V be the set of all pairs (x,y) of real numbers with the addition + and scalar multiplication* defined by:
    (x1,y1)+(x2,y2)= (x1 + x2 , y1+y2) and c*(x,y)=(x,cy)

    Show that V with the above operation is not a vector space. Find at least one axiom that fails and give an example showing that the axiom fails!!!..
    ***Let α = (x, y)
    Then for real numbers a and b we have
    (a + b) α = (a + b) (x, y)
    = ( x, (a+b)y )

    Now aα = a(x, y)
    = (x, ay)
    bα = b(x, y)
    = (x, by)
    and aα + bα = (x, ay) + (x, by)
    = ( 2x , ay + by)
    [IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image001.gif[/IMG] (a + b) α
    Therefore, V is not a vector space.

    thanks
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  2. #2
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by cris123 View Post
    i did the work just i need more details and justify by stating axions and theorems,, please help

    2) Let V be the set of all pairs (x,y) of real numbers with the addition + and scalar multiplication* defined by:
    (x1,y1)+(x2,y2)= (x1 + x2 , y1+y2) and c*(x,y)=(x,cy)

    Show that V with the above operation is not a vector space. Find at least one axiom that fails and give an example showing that the axiom fails!!!..
    ***Let α = (x, y)
    Then for real numbers a and b we have
    (a + b) α = (a + b) (x, y)
    = ( x, (a+b)y )

    Now aα = a(x, y)
    = (x, ay)
    bα = b(x, y)
    = (x, by)
    and aα + bα = (x, ay) + (x, by)
    = ( 2x , ay + by)
    [IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image001.gif[/IMG] (a + b) α
    Therefore, V is not a vector space.

    thanks
    what is your problem then? as i see it, there is nothing wrong with your proof..
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