1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Orthogonal Matrices and span

  1. Jan 17, 2009 #1
    1. The problem statement, all variables and given/known data

    If w is orthogonal to u and v, then show that w is also orthogonal to span ( u , v )

    2. Relevant equations

    two orthogonal vectors have a dot product equalling zero

    3. The attempt at a solution

    I can see this geometrically in my mind, and I know that w . u = 0 and w . v = 0
    but I don't know or understand how I can show this for its span in writing.
  2. jcsd
  3. Jan 17, 2009 #2
    a vector in span(u,v) is of the form au+bv. So w . (au+bv) = 0, using the distributivity of the dot product.
  4. Jan 17, 2009 #3
    you said that span(u,v) is in this form au+bv

    w . (au+bv) = 0
    w . au + w . bv = 0

    where a and b are any scalar numbers
    and that's all? There's no more to it?

    thanks grief. That one little bit helped a lot!
    Last edited: Jan 17, 2009
  5. Jan 17, 2009 #4
    No, Grief said a vector in the subspace span(u,v) is of the form au+bv for scalars a and b. To say that a vector is orthogonal to a subspace means that the vector is orthogonal to each vector in that subspace.

    You need to show that given a vector x in span(u,v), we have w.x=0 . From above, x is of the form au+bv, so you want to show that w.(au+bv)=0. This means beginning with w.(au+bv) and showing it equals 0. As Grief already said, to do so just requires distributivity and recognising that w being orthogonal to u and to v means that w.u=0 and w.v=0.
  6. Jan 18, 2009 #5
    To make sure I got this right one more time:
    a vector in span(u,v) is in this form au+bv

    making (au+bv) dot w = 0 shows it is orthogonal, meaning any vector in that span(u,v) is orthogonal to w
    then it'll become w . au + w . bv = 0
    then w . au = a(w.u) = 0 since w.u = 0 (orthogonal)
    w . bv = 0 since w . v = 0 (orthogonal)
    so then 0 = 0

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook