Results 1 to 3 of 3

Math Help - Need help with Extension Fields

  1. #1
    Junior Member
    Joined
    Sep 2008
    From
    Indiana
    Posts
    26

    Need help with Extension Fields

    Prove that if V is a finite-dimensional vector space over a field F, then a subset { \beta_{1}, \beta_{2}, ..., \beta_{n}} of V is a basis for V over F if and only if every vector in V can be expressed uniquely as a linear combination of the \beta_{i}.

    First things first. Could someone explain the concept of a linear combination? Wikipedia and my book aren't quite cutting it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    If you haven't grasped the idea of a linear combination, I highly doubt you understand what a basis is.

    What is it that you don't understand? Where do you get stuck reading the wikipedia article?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2008
    From
    Indiana
    Posts
    26
    What I've gathered from Wikipedia is that a linear combination is a set of scalars from a field multiplied by a set of vectors from a vector space. So in the literal sense of knowing the definition and knowing that a linear combination is the elements of a field multiplied by a the vectors in a vector space, I guess I understand it.

    I don't understand what that does for us. And I have a rough thought of what a basis is, but probably not a well enough working idea to understand why that statement is true.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Extension fields
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 16th 2011, 07:47 AM
  2. Elements of Extension Fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 19th 2011, 05:38 PM
  3. Extension Fields
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 22nd 2011, 06:41 PM
  4. Extension fields
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 19th 2010, 07:15 PM
  5. Extension fields / splitting fields proof...
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2007, 07:29 AM

Search Tags


/mathhelpforum @mathhelpforum