Direct sum basis ($((u_1,0),\dots ,(u_p,0),(0,v_1),\dots ,(0,v_q))$)

For bases $(u_i)$ of $U$ and $(v_j)$ of $W$, the displayed pairs form a basis of $U\oplus W$. Expanding each component proves spanning; comparing components of a vanishing linear combination proves linear independence. Therefore $\dim (U\oplus W)=\dim U+\dim W$ for two finite-dimensional vector spaces.