= Addition map of subspaces
{title2=$g:U\times W\to U+W,\quad (u,w)\mapsto u+w$}
For <vector subspaces> $U,W$ of a common <vector space>, this is a <surjective linear map>. Its <kernel of a linear map> is $\{(x,-x):x\in U\cap W\}$, which has a <linear isomorphism> to $U\cap W$ by $x\mapsto(x,-x)$. The <rank-nullity theorem> and the <direct sum basis> give the <dimension formula for a sum of subspaces>. The paired domain keeps the two contributions separate even when their underlying <vectors> overlap.
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