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Homework Help: Basis of sum/union of subspaces

  1. Jan 31, 2009 #1
    1. The problem statement, all variables and given/known data
    I need to find the basis of W1 + W2 and W1 intersect W2

    (It's part of a larger homework problem that I know how to do, but I am stuck on the trivial step...per usual)

    [tex]W_1 = \left(\begin{array}{c c} x & -x \\ y & z \end{array}\right)[/tex]

    [tex]W_2 = \left(\begin{array}{c c} a & b \\ -a & c \end{array}\right)[/tex]


    2. Relevant equations
    Dim(A) + Dim(B) = Dim(A intersect B) + Dim(A + B)


    3. The attempt at a solution

    [tex]\alpha \in W_1, \alpha = x \left(\begin{array}{c c} 1 & -1 \\ 0 & 0 \end{array}\right) + y \left(\begin{array}{c c} 0 & 0 \\ 1 & 0 \end{array}\right) + z \left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)[/tex]

    [tex]\beta \in W_2, \beta = a \left(\begin{array}{c c} 1 & 0 \\ -1& 0 \end{array}\right) + b \left(\begin{array}{c c} 0 & 1 \\ 0 & 0 \end{array}\right) + c \left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)[/tex]

    So the dimensions of each subspace of those subspaces is 3.

    Obviously, [tex]\left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)[/tex] is in both sets (and thus in the intersection).

    I'm having a hard time finding the other vector in the basis.

    Is [tex]\left(\begin{array}{c c} 1 & -1 \\ -1 & 0 \end{array}\right)[/tex] in the basis set of vectors of the intersection?
     
  2. jcsd
  3. Jan 31, 2009 #2
    consider W_1+W_2.

    you'll get

    [tex]W_1+W_2 = \left(\begin{array}{c c} x+a & -x+b \\ y-a & z+c \end{array}\right)[/tex]

    what would a basis for that be? and then its dimension?

    also, keep in mind the equation you posted, and think about what you said about [tex]
    \left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)
    [/tex] being in the intersection.

    what exactly does the intersection look like?
     
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