= Rank bounds for a sum of linear maps
{title2=$|\operatorname{rank}A-\operatorname{rank}B|\leq\operatorname{rank}(A+B)\leq\operatorname{rank}A+\operatorname{rank}B$}
The image of $A+B$ lies in $\operatorname{im}A+\operatorname{im}B$. The <dimension formula for a sum of subspaces> gives the upper bound. Applying it to $A=(A+B)-B$ and then interchanging $A,B$ gives the lower bound. Both extremes are sharp: cancelling diagonal blocks gives the difference of ranks; diagonal maps with supports covering the whole space and positive coefficients give full rank when the sum of prescribed ranks is at least the dimension.
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