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descendency

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## Homework Statement

I need to find the basis of W

_{1}+ W

_{2}and W

_{1}intersect W

_{2}

(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?