Results 1 to 4 of 4

Math Help - Proving a Vector Space

  1. #1
    Junior Member
    Joined
    Sep 2007
    Posts
    29

    Proving a Vector Space

    I just can't completely answer this question, does it want me to prove it using all 10 axioms? Or is there another way



    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Here is an easier way to do this without using all 10 properties of a vector space.

    Theorem: Let V be a vector space over a field (you probably learned it over \mathbb{R}). Let W be a subset of V so that \bold{a}+\bold{b} \in W for all \bold{a},\bold{b} \in W and c\bold{a} \in W for all \bold{a}\in W and c\in \mathbb{R}. Then W is a vector space over the field ( \mathbb{R}).

    So you only need to check closure of vector addition and scalar multiplication.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2007
    Posts
    29
    I'm not sure if I'm correct but, the closure under addition wouldn't be true because the final matrix has to end up as

    [a+c 2]
    [ 2 b+d]
    ???

    I'm almost positive axiom 6 for scalar multiplication holds, but I'm not sure about axiom 1, if you could tell me whether my above theory is correct or not that would be great. I am not very good at proving axioms at least 1 and 6, because there is no left and right hand side of the equation to compare, so I get really lost.
    Last edited by Adrian; October 29th 2007 at 10:31 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by Adrian View Post
    I'm not sure if I'm correct but, the closure under addition wouldn't be true because the final matrix has to end up as

    [a+c 2]
    [ 2 b+d]
    ???

    I'm almost positive axiom 6 for scalar multiplication holds, but I'm not sure about axiom 1, if you could tell me whether my above theory is correct or not that would be great. I am not very good at proving axioms at least 1 and 6, because there is no left and right hand side of the equation to compare, so I get really lost.
    no, that is not the way the definition of addition gives you.. the definition says that you will just add the entries (1,1) and (2,2), while entries (1,2) and (2,1) are fixed to 1.
    also, the same thing with closure of multiplication..
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Banach space with infinite vector space basis?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 24th 2011, 07:23 PM
  2. Proving vector space properties
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 27th 2010, 04:53 AM
  3. Replies: 15
    Last Post: July 23rd 2010, 12:46 PM
  4. Isomorphism beetwenn vector space and sub space
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 30th 2008, 11:05 AM
  5. Proving a vector space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 20th 2008, 10:22 PM

Search Tags


/mathhelpforum @mathhelpforum