- #1
descendency
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Homework Statement
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]
Homework Equations
Dim(A) + Dim(B) = Dim(A intersect B) + Dim(A + B)
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?