1. Not finding help here? Sign up for a free 30min 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!

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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Basis of sum/union of subspaces
  1. Basis of a subspace? (Replies: 4)

  2. Basis for subspace (Replies: 8)

  3. Basis of a subspace (Replies: 1)

  4. Subspace and basis (Replies: 8)

  5. Basis for a subspace (Replies: 6)

Loading...