Results 1 to 7 of 7

Math Help - dimention of group question

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    dimention of group question

    there is
    v1,..,vk,u,w vectors on space V
    the group {v1,..,vk,u} is independant
    and
    Sp\{v1,..,vk\}\cap Sp\{u,w\}\neq\{0\}
    find the dimention of sp{v1..vk,u,w}
    ?
    if {v1,..,vk,u} is independant then its span dimention is k+1
    and span so {v1,..,vk} independat to and dim sp{v1,..,vk}=k
    the dimention of sp{v1..vk,u,w} is
    dim(sp{v1..vk,u,w})=dim(sp{v1..vk}and sp{u,w}})=dim(sp{v1..vk}+sp{u,w})=dim(sp{v1..vk})+ dim(sp{u,w})-dim(sp{v1..vk} intersect sp{u,w}})
    if v1..vk,u is independant then v1,..,vk is independant too so dim (sp{v1..vk})=k
    now regardingthe other two members i dont know
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2008
    Posts
    410

    Re: dimention of group question

    Let x be a nonzero vector in

    \text{span}\{v_1,\cdots,v_k\}\cap\text{span}\{u,w\  }.

    Then there are constants a_1,\cdots,a_k and b,c such that a_1v_1+\cdots+a_kv_k=x=bu+cw and hence

    w=c^{-1}a_1v_1+\cdots+c^{-1}a_kv_k-c^{-1}bu.

    So w\in\text{span}\{v_1,\cdots,v_k,u\} which means \dim\text{span}\{v_1,\cdots,v_k,u,w\}=\dim\text{sp  an}\{v_1,\cdots,v_k,u\}=k+1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    Re: dimention of group question

    ok so we need to find
    dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})
    dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})- dim(sp{v1,..,vk,u}\cap sp{w})=k+1
    beacuse dim(sp({v1,..,vk,u}}) \cap sp{w})=0
    so
    dim(sp{v1,..vk,u,w})=k+1
    correct?
    Last edited by transgalactic; June 22nd 2011 at 12:43 PM. Reason: fdhjf
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Nov 2010
    Posts
    193

    Re: dimention of group question

    Quote Originally Posted by transgalactic View Post
    ok so we need to find
    dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})
    dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})- dim(sp{v1,..,vk,u}\cap sp{w})=k+1
    beacuse dim(sp{v1,..,vk,u}\cap sp{w})=0
    so
    dim(sp{v1,..vk,u,w})=k+1
    correct?
    This is extremely unreadable. When you write out proofs in mathematics, you need to make sure that everyone can understand what you are trying to say. We aren't in your head; all we see is what you have written down, so if we can't interpret that, then we are out of luck.

    Anyway, hatsoff has given the complete proof, with no steps missing.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    Re: dimention of group question

    hatsoff proved a very important step
    it showed that i divided the group in the wring way

    i made some mistakes in latex
    but its only about one formula for the dimention of the sum of spans

    here is the edited:
    ok so we need to find

    dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})

    dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})- dim(sp{v1,..,vk,u}\cap sp{w})=k+1

    beacuse dim(sp({v1,..,vk,u}}) \cap sp{w})=0

    so

    dim(sp{v1,..vk,u,w})=k+1

    correct?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Nov 2010
    Posts
    193

    Re: dimention of group question

    Quote Originally Posted by transgalactic View Post
    hatsoff proved a very important step
    it showed that i divided the group in the wring way

    i made some mistakes in latex
    but its only about one formula for the dimention of the sum of spans

    here is the edited:
    ok so we need to find

    dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})

    dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})- dim(sp{v1,..,vk,u}\cap sp{w})=k+1

    beacuse dim(sp({v1,..,vk,u}}) \cap sp{w})=0

    so

    dim(sp{v1,..vk,u,w})=k+1

    correct?
    1. How do you get dim(sp({v1,..,vk,u}}) \cap sp{w})=0?

    2. If (1) were true, then

    dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})- dim(sp{v1,..,vk,u}\cap sp{w})=dim(sp{v1,..,vk,u})+dim(sp{w})-0=(k+1)+1-0=k+2, NOT k+1.

    I'll say again: hatsoff has already given the COMPLETE proof.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    Re: dimention of group question

    oohh yes i got it
    soryy i am a little slow on the comprehetion side in this hour

    thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. dimention of kernel
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 29th 2011, 09:33 AM
  2. dimention laws
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: June 29th 2011, 05:39 AM
  3. A question about Group
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 26th 2010, 01:36 AM
  4. Group Theory Question, Dihedral Group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 4th 2008, 10:36 AM
  5. One more group Question
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 18th 2007, 04:16 AM

Search Tags


/mathhelpforum @mathhelpforum