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Linear algebra

  1. Jan 26, 2009 #1
    in this question i am given 2 subspaces of R4



    and am asked to find
    1] a basis and dimention for W
    2] a basis for W+U
    3] a basis for W[tex]\cap[/tex]U

    since i only have limitations on x1 and x4 i call x2=t x3=s x1=x4=q

    therefore W=sp{(1,0,0,1),(0,1,0,0),(0,0,1,0)} and dimW=3

    to find a basis for W+U, i look for linearly independant vectors that span the space, so i set up a matrix to see which are combinations of the others,
    i got W+U=sp{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}

    these 4 span the whole of R4 so does this mean that W+U = R4
    to find a basis of W[tex]\cap[/tex]U, i found a homogenic system for each subspace and compared the 2.
    taking a random vector (a b c d) in the subspace, i get
    for W==> -4a+b+3c-sd=0
    for U==> a-d=0
    from the combination i get -6a+b+3c=0
    a=t b=6t-3s c=s d=t

    and so i get W[tex]\cap[/tex]U=sp{(1,6,0,1)(0,-3,1,0)} dim=2

    does this all look okay,
    also, if i am asked to find a basis and i write the span, is it the same thing, or must i write just one possible basis in the span
    Last edited: Jan 27, 2009
  2. jcsd
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