Originally Posted by

**hatsoff** I don't think that is true (and thus can't be proven). I'm rusty on subspaces but...

Let $\displaystyle V=R^3$...

Let $\displaystyle W_1$ be the set spanned by $\displaystyle \left[\begin{array}{c}0\\0\\1\end{array}\right]$. Let $\displaystyle W_2$ be the set spanned by $\displaystyle \left[\begin{array}{c}0\\1\\0\end{array}\right]$. Then $\displaystyle W_1\cup W_2$ is the set spanned by $\displaystyle \left[\begin{array}{c}0\\1\\1\end{array}\right]$. So $\displaystyle W_1$, $\displaystyle W_2$ and $\displaystyle W_1\cup W_2$ are subspaces of $\displaystyle V$, but $\displaystyle W_1$ and $\displaystyle W_2$ are non-overlapping.