= Solution
Apply the preceding <rank bounds for a sum of linear maps> to $\alpha=(\alpha+\beta)+(-\beta)$. Since multiplying a <linear map> by $-1$ does not change its image, $\operatorname{rank}\alpha\leq\operatorname{rank}(\alpha+\beta)+\operatorname{rank}\beta$. Thus $\operatorname{rank}(\alpha+\beta)\geq\operatorname{rank}\alpha-\operatorname{rank}\beta$. Interchanging the maps gives the other difference, proving
$$
\boxed{\operatorname{rank}(\alpha+\beta)\geq|\operatorname{rank}\alpha-\operatorname{rank}\beta|.}
$$
Back to article page
