Solution
= Solution
\b[False.] Take $\alpha(x,y)=x$, $X=\operatorname{span}\{(1,0)\}$, 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 $X\cap Y=Y\ne\{0\}$.