Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof / 6 / b / i / Solution

The intersection of vector subspaces $W_1\cap W_2$ contains zero, so $(0,0)\in U$. If $(u,-u),(v,-v)\in U$ and $a,b$ are scalars, thenbecause $au+bv\in W_1\cap W_2$ by the vector subspace conditions. These pairs belong to $W_1\times W_2$, since both $u$ and $-u$ belong to the required factors. Thus $U$ is a vector subspace of $W_1\times W_2$.