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Subspaces and basis

  1. Jan 22, 2009 #1
    i am given these 2 groups
    W=sp{(1 0 2 0) (1 1 1 1) (1 0 0 0)}
    U=sp{(1 0 1 1) (1 2 1 2) (0 0 1 0)}

    and am asked to find
    a basis for each one and their dimention
    a basis for W+U
    a basis for W[tex]\cap[/tex]U
    -----------------------------------------------
    for the basis i found that they are both linearly independant therefore my basis is the span given and the dimention is 3 for both of them
    ------------------------------------------------
    how do i find a basis for W+U? can i take the 6 vectors given and check which are dependant and which are independant, take the independant ones in which case i get

    w+u=sp{(1 0 2 0) (1 1 1 1) (1 0 0 0) (1 0 1 1)}
    dim(W+U)=4
    ------------------------------------------------
    for W[tex]\cap[/tex]U am i looking for all the vectors in W which are perpendicular to U? how would i do this?
    i know how to find one vector perpendicular to a subspace but how do i find a basis for a group perpendicular to another group
     
  2. jcsd
  3. Jan 22, 2009 #2
    for vectors in W which intersect U can i set up a parameter vector (a b c d) then compare it to the basis of W to get a homogenic system

    1 1 1 | a
    0 1 0 | b
    2 1 0 | c
    0 1 0 | d

    then i get d-b=0

    then do the same for u

    1 1 0 | a
    0 2 0 | b
    1 1 1 | c
    1 2 0 | d

    and i get b-2(d-a)=0

    then any vector that adheres to these 2 conditions is an answer.
     
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