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Linear Algebra: Geometric Description of Span

  1. Aug 25, 2007 #1
    1. The problem statement, all variables and given/known data
    Give a geometric description of the Span {[tex]V_{1}[/tex],[tex]V_{2}[/tex]} for the vectors [tex]V_{1}[/tex] = [8, 2, -6]
    and [tex]V_{2}[/tex] = [12, 3, -9]

    Those should be columns but I couldn't figure that out in latex, sorry.

    2. The attempt at a solution

    I have a solution, what I need help with is understanding it, this Span stuff just isn't clicking.

    [tex]V_{2}[/tex] = (3/2)[tex]V_{1}[/tex]

    So, a[tex]V_{1}[/tex] + b[tex]V_{2}[/tex] = a[tex]V_{1}[/tex] +b(3/2)[tex]V_{2}[/tex] = (a + (3b/2))[tex]V_{1}[/tex]

    So Span {[tex]V_{1}[/tex], [tex]V_{2}[/tex]} is the set of points on the line through [tex]V_{1}[/tex] and 0.

    As I said I know the solution, but I don't understand it. I've reread the section on Span many times and I still don't know what exactly is going on. Any advice or help would be much appreciated.
  2. jcsd
  3. Aug 25, 2007 #2


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    You seem to understand it completely. The span is the subspace consisting of all vectors that are linear combinations of V1 and V2. And this subspace is the line through 0 and V1. There is not much I can add.
    Last edited: Aug 25, 2007
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