Results 1 to 3 of 3

Math Help - Basis and Dependent Vector - Linear Algebra

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    78

    Basis and Dependent Vector - Linear Algebra

    Suppose that B is a basis for the vector space V and ~v is not an element of B.
    If we make a new collection C by adding ~v to B, that is
    C = B U {~v};
    show that C must be a dependent set of vectors.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by flaming View Post
    Suppose that B is a basis for the vector space V and ~v is not an element of B.
    If we make a new collection C by adding ~v to B, that is
    C = B U {~v};
    show that C must be a dependent set of vectors.
    Because we can express ~v in terms of linear combinations of B.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2005
    From
    Earth
    Posts
    1,599
    Just to reword TPH's idea...

    So the basis of vector space V is a subset that can represent any vector in V as a linear combination of vectors from B. In order for a set to be linear independent, no linear combination of vectors in that set can produce 0 (other than the trivial case where all the coefficients are 0).

    Thus since C contains ~v, not in B, but in V, we can represent ~v as a linear combination of vectors in the basis, which means we could represent -(~v), and the sum of the linear combinations could now be zero non-trivially.

    How bad did I mess up TPH?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear Algebra- Basis
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 18th 2009, 10:04 AM
  2. linear algebra- vector-basis-subspace
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: October 10th 2008, 07:23 PM
  3. Help with Linear Algebra , Orthonormal Basis
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: September 14th 2008, 08:41 PM
  4. Basis in Linear Algebra
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 10th 2007, 11:48 PM
  5. Linear algebra find the basis
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 19th 2007, 07:06 PM

Search Tags


/mathhelpforum @mathhelpforum