Results 1 to 4 of 4

Thread: How to determine whether a set spans in Rn

  1. #1
    Junior Member
    Join Date
    Sep 2007
    Posts
    97

    How to determine whether a set spans in Rn

    In general, I'd like to know how to determine whether a set of m vectors spans in Rn. Do I have to look at the rank of the matrix that the vectors form?

    Like if 3 vectors in R3, have a rank of 3, does this mean they span in R3?

  2. #2
    Elite Member
    Join Date
    Sep 2005
    Posts
    7,293

    Re: How to determine whether a set spans in Rn

    A set of vectors spans if they can be expressed as linear combinations. Say we have a set of vectors we can call S in some vector space we can call V. The subspace, we can call W, that consists of all linear combinations of the vectors in S is called the spanning space and we say the vectors span W.

    Here is an example of vectors in R^3.

    Say we have [tex]V_{1}=(1,1,2), \;\ v_{2}=(1,0,1), \;\ v_{3}=(2,1,3)[/tex]

    We want to see if they span or not.

    We have to find whether an arbitrary vector, say, [tex]b=(b_{1},b_{2},b_{3})[/tex] can be expressed as a linear combo [tex]b=k_{1}v_{1}+k_{2}v_{2}+k_{3}v_{3}[/tex] of the vectors [tex]v_{1},v_{2},v_{3}[/tex].

    Set up a system of equations in terms of the components:

    [tex](b_{1},b_{2},b_{3})=k_{1}(1,1,2)+k_{2}(1,0,1)+k_{3 }(2,1,3)[/tex]

    [tex](b_{1},b_{2},b_{3})=(k_{1}+k_{2}+2k_{3}, \;\ k_{1}+k_{3}, \;\ 2k_{1}+k_{2}+3k_{3})[/tex]

    [tex]k_{1}+k_{2}+2k_{3}=b_{1}[/tex]

    [tex]k_{1} \;\ \;\ +k_{3}=b_{2}[/tex]

    [tex]2k_{1}+k_{2}+3k_{3}=b_{3}[/tex]

    The system is consistent for all [tex]b_{1},b_{2},b_{3}[/tex] iff the matrix of coefficients:

    [tex]A=\begin{bmatrix}1&1&2\\1&0&1\\2&1&3\end{bmatrix}[/tex]

    has a determinant that is not equal to 0.

    But this determinant does equal 0, so it DOES NOT span.

  3. #3
    Junior Member
    Join Date
    Sep 2007
    Posts
    97

    Re: How to determine whether a set spans in Rn

    Thank you for the clarification.

  4. #4
    Full Member
    Join Date
    Mar 2009
    Posts
    259

    Re: How to determine whether a set spans in Rn

    Quote Originally Posted by Idealistic
    In general, I'd like to know how to determine whether a set of m vectors spans in Rn. Do I have to look at the rank of the matrix that the vectors form?

    Like if 3 vectors in R3, have a rank of 3, does this mean they span in R3?
    I wouldn't want to say you had to look at the rank, but that will certainly do.

    The columns - or rows - of a rank r matrix will span an r-dimensional space. If r=3 and the vectors are in R^3, then this must be the whole space.

    However, that's not the only way to do it. For example, you could look at the null space, and use the rank-nullity theorem.
    -------
    Function Integrator for iPhone - definite integrals in your pocket - http://bit.ly/lpzjZ6

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •