The original question is "Consider subspaces W1 = {(x,y,z) | x - y = 0} and W2 = {(x,y,z) | x + z = 0}. Show that $R_{3} = W_{1} + W_{2}$ but $R_{3} \neq W_{1} \bigoplus W_{2}$"
To do this, we'll have to show $W_{1} \bigcap W_{2} \neq \{0\}$ and we get that by solving the system so we get (x,y,z) = s(-1, -1, 1) but the part that I don't understand in the solutions given is that it says "so $W_{1} \bigcap W_{2} =$ span{(-1,-1,1)} and hence $W_{1} \bigcap W_{2} \neq \{0\}$".
My main source of confusion: How does the intersection of the two subspaces equate to the span{(-1, -1, 1)}?
The reason why it is equal to $\mathrm{Span}\{(-1, -1, 1)\}$ is in the fact that there are more than one vector in the intersection of this two subspaces. This is true, because you can take vector $(-1, -1, 1)$ and multiply it by any constant and the resulting vector will also be in the intersection, and set $\{\alpha (-1, -1, 1)\ |\ \alpha \in F\}$ is equal to $\mathrm{Span}\{(-1, -1, 1)\}$ by definition. Also note that intersection of two subspaces is always a subspace, so it should contain all vectors proportional to $(-1, -1, 1)$.