\b[False.] Take $U=\mathbb R^2$, $V=\mathbb R$, $\alpha(x,y)=x+y$, $X=\operatorname{span}\{(1,0)\}$, and $Y=\operatorname{span}\{(0,1)\}$. 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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 1 / 9F / iv / Solution

 Solution (source code)