Subadditivity of matrix rank
= Subadditivity of matrix rank
{title2=$\operatorname{rank}(A+B)\leq\operatorname{rank}A+\operatorname{rank}B$}
The image of the <linear map> $A+B$ is contained in the sum of the images of $A$ and $B$. Since the <dimension of a vector space> of this sum is at most the sum of their dimensions, <matrix rank> is subadditive. Iteration shows that a sum of $r$ <matrices> of <matrix rank> at most one has <matrix rank> at most $r$, over any <field>.