Linear Algebra Done Right · Problem 8A.9

This problem asks you to connect two seemingly opposite properties of powers of a linear operator: when the kernel stops growing and when the range stops shrinking. The key insight is that $\operatorname{null} T^m$ and $\operatorname{range} T^m$ are dual notions—one measures what $T^m$ "kills," the other what it "reaches"—and they're constrained by the rank-nullity theorem applied repeatedly. If the null space stabilizes at step $m$, the rank can't grow further, so the range must also stabilize; the converse follows by the same dimension-counting argument.

This problem asks you to connect two seemingly opposite properties of powers of a linear operator: when the kernel stops growing and when the range stops shrinking. The key insight is that $\operatorname{null} T^m$ and $\operatorname{range} T^m$ are dual notions—one measures what $T^m$ "kills," the other what it "reaches"—and they're constrained by the rank-nullity theorem applied repeatedly. If the null space stabilizes at step $m$, the rank can't grow further, so the range must also stabilize; the converse follows by the same dimension-counting argument.