Direct sum basis
= Direct sum basis
{title2=$((u_1,0),\ldots,(u_p,0),(0,v_1),\ldots,(0,v_q))$}
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>.