Results 1 to 2 of 2

Math Help - Vector Space and Basis

  1. #1
    Newbie
    Joined
    Sep 2008
    Posts
    23

    Vector Space and Basis

    Let S = {v1, v2,...vn} be a set of nonzero vectors in a vector space V such that every vector in V, can be written in one and only one way as a linear combination of the vectors in S. Prove that S is a basis for V.

    What I have so far is that if every vector in V can be written as a linear combination of the vectors in S, then S spans V. If S spans V then

    v = a1v1 + a2v2 + ... + anvn

    Now all I would have to do is prove that it is linearly independent right? I'm not sure how to go on after this.

    Would I have to find a v = b1v1 +...+ bnvn so that v - v = 0? That would show the linear independence. That means a1 = b1 ... an = bn. I'm not sure if I'm going about this in a backwards 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
    Quote Originally Posted by Brokescholar View Post
    Let S = {v1, v2,...vn} be a set of nonzero vectors in a vector space V such that every vector in V, can be written in one and only one way as a linear combination of the vectors in S. Prove that S is a basis for V.
    Notice that S definitely spams V because the hypothesis says every vector in V is a linear combination of vectors in set S. It remains to prove that S is a linearly independent set of vectors. Say that k_1\bold{v}_1 + ... + k_n \bold{v}_n = \bold{0}. But \bold{0} = 0\bold{v}_1 + ... + 0 \bold{v}_n. Therefore by uniquness it follows that k_1=0, ... , k_n = 0. Thus, S is linearly independent.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vector Space, Basis
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 3rd 2011, 09:50 AM
  2. Basis of ker L --> Basis of vector space?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: September 17th 2011, 09:57 AM
  3. Banach space with infinite vector space basis?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 24th 2011, 07:23 PM
  4. vector space/basis
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 2nd 2010, 04:41 PM
  5. Vector Space Basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 26th 2008, 02:49 PM

Search Tags


/mathhelpforum @mathhelpforum