I was asked to give examples of $3$ subspaces where $W + V + U$ is not the direct sum of these $3$ subspaces. $W$, $V$, and $U$ are subspaces of a vector space, just to clarify.
I am having trouble finding out whether or not the intersection of the vectors is the zero vector or not. For example, if i take the $3$ subspaces $W = (1,0,1)$, $V = (1,1,0)$, $U = (2,3,1)$, then $W + U + V$ is $(3,4,2)$. Would this be also be a direct sum? or not? I dont think it would. How do i check that their intersection is not the zero vector?
But if i took for example $W = (1,0,1)$, $V = (0, 6, 0)$, this would be a direct sum right?
What if i took $W = (1,0,1)$, $V = (0, 6, 0)$ and $U = (2, 3, 2)$, would this be a direct sum or not? Is it because i could write $U$ as $0.5 V + 2W$ ? Does the direct sum have to do with linear dependance?
Thanks!