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Homework Help: Orthagonal vectors/vector spaces

  1. Jan 20, 2009 #1
    givem,-W is the group of all the vectors in R4 which are orthagonal to the vectors v1(1 0 1 0) and also v2(1 2 1 1)
    a) find a basis and the dimention of W
    b) find, in W, a vector which is orthagonal to the vector (1 -2 1 3)

    a)

    i call this vector w=(a b c d) and say
    v1(dot)w=0
    v2(dot)w=0

    therefore

    a+0b+c+0d=0 a=-c
    a+2b+c+d=0 d=-2b

    i will have 2 free parameters here, 4 unknows -rank2=2
    a=t
    d=u

    so w=(t, -2u, -t, u)

    W=sp{(1 0 -1 0), (0 -2 0 1)}
    dim(w)=2
    ----------------------------
    b)

    (t, -2u, -t, u) dot (1 -2 1 3)=0

    t+4u-t+3u=0
    7u=0
    u=0

    therefore any vector that fits
    (t 0 -t 0) is orthagonal to (1 -2 1 3) and a part of W


    is this all correct
     
  2. jcsd
  3. Jan 20, 2009 #2

    Mark44

    Staff: Mentor

    Looks good.
     
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