I don't understand how if U and W are subspaces of V such that $\displaystyle S\left ( \vec{u} \right ) \epsilon W,\forall \vec{u}\epsilon U$ and $\displaystyle S\left ( \vec{w} \right ) \epsilon U,\forall \vec{w}\epsilon W$ then U + W will be S-invariant even if U and W aren't. Can someone explain this to me? I need to understand this to make an operator S for 2x2 matrices.