 Solution

False. Take $U={\mathbb{R}}^2$, $V=\mathbb{R}$, $\alpha (x,y)=x+y$, $X=\mathrm{span}\{(1,0)\}$, and $Y=\mathrm{span}\{(0,1)\}$. Then $U=X\oplus Y$ is a direct sum, but $\alpha (X)=\alpha (Y)=\mathbb{R}$, so their intersection is not zero. Their ordinary sum is $V$, as always follows from surjective function and $U=X+Y$; the failure is precisely the direct sum assertion.