# Basis of sum/union of subspaces

1. Jan 31, 2009

### descendency

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)

$$W_1 = \left(\begin{array}{c c} x & -x \\ y & z \end{array}\right)$$

$$W_2 = \left(\begin{array}{c c} a & b \\ -a & c \end{array}\right)$$

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

3. The attempt at a solution

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

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

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

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

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

Is $$\left(\begin{array}{c c} 1 & -1 \\ -1 & 0 \end{array}\right)$$ in the basis set of vectors of the intersection?

2. Jan 31, 2009

### jimmypoopins

consider W_1+W_2.

you'll get

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

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 $$\left(\begin{array}{c c} 0 & 0 \\ 0 & 1 \end{array}\right)$$ being in the intersection.

what exactly does the intersection look like?