Results 1 to 3 of 3

Math Help - Is this a vector space?

  1. #1
    Junior Member
    Joined
    May 2008
    Posts
    42

    Is this a vector space?

    Let (a,b) and (c,d) be in V and c be in R, we define:
    (a,b) + (c,d) = (a+2c,b+3d) and c(a,b) = (ca,cb)

    Using the eight properties.
    I know that the multiplication is normal, so those properties work, but the addition does not hold under the conditions, does it?

    (u + v) + w =/= u + ( v + w)
    u + v =/= v + u
    -> If u = (a,b) and v = (c,d),
    u + v = (a,b) + (c,d) = (a+2c,b+3d)
    v + u = (c,d) + (a,b) = (c+2a,d+3b)

    So it's not a vector space. Is this correct ??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,963
    Thanks
    1631
    Quote Originally Posted by Volcanicrain View Post
    Let (a,b) and (c,d) be in V and c be in R, we define:
    (a,b) + (c,d) = (a+2c,b+3d) and c(a,b) = (ca,cb)

    Using the eight properties.
    I know that the multiplication is normal, so those properties work, but the addition does not hold under the conditions, does it?

    (u + v) + w =/= u + ( v + w)
    You should show specifically: if u= (a,b), v= (c,d), and w= (e, f) then u+ v= (a+2c, b+3d) and then (u+ v)+ w= (a+2c+ 2u, b+3d+ 3f) while v+w= (c+2e, d+ 3f) and u+ (v+w)= (a+ 2(c+3e),b+3(d+3f))= (a+2c+ 6e, b+3d+ 9f) which is NOT the same as (a+2c+2u, b+3d+3f).

    u + v =/= v + u
    -> If u = (a,b) and v = (c,d),
    u + v = (a,b) + (c,d) = (a+2c,b+3d)
    v + u = (c,d) + (a,b) = (c+2a,d+3b)
    Okay, here you did exactly that.

    So it's not a vector space. Is this correct ??
    Yes. In fact, since all properties have to be true you really only needed to show that u+ v\ne v+ u.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    May 2008
    Posts
    42
    Thank you very much!
    I did the (u+v)+w = u+(v+w) out on paper as you did only to realize the other one was shorter haha.
    Thank you very much for answering my question, you are very clear
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Dual Space of a Vector Space Question
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 16th 2011, 03:02 AM
  2. Replies: 2
    Last Post: April 1st 2011, 02:40 AM
  3. Banach space with infinite vector space basis?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 24th 2011, 06:23 PM
  4. Replies: 15
    Last Post: July 23rd 2010, 11:46 AM
  5. Isomorphism beetwenn vector space and sub space
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 30th 2008, 10:05 AM

Search Tags


/mathhelpforum @mathhelpforum