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 / a

Writing the given bases as $(u_1,\dots ,u_p)$ and $(v_1,\dots ,v_q)$, a basis of the Cartesian product with its componentwise vector space operations isThis is the direct sum basis. Therefore $\dim (W_1\times W_2)=p+q=\dim W_1+\dim W_2$. The construction includes zero-dimensional factors, whose corresponding portion of the basis is empty.