 Solution

False. Take $\alpha (x,y)=x$, $X=\mathrm{span}\{(1,0)\}$, and $Y=\{0\}$. Then $V=\alpha (X)\oplus \alpha (Y)=\mathbb{R}\oplus \{0\}$, but $X+Y\ne U$ because $(0,1)$ is missing. Thus the premise does not force spanning. It need not force zero intersection either: with the same $\alpha$, take $X=U$ and $Y=\ker \alpha$. Their images still have a direct sum, while $X\cap Y=Y\ne \{0\}$.