Kunze Linear Algebra · Problem 7.2.2

The core idea is understanding when a subspace can be "split off" cleanly by the operator itself. A $T$-invariant complement to $R$ means we can decompose $V = R \oplus W$ where $T$ maps $W$ into itself—this is possible precisely when $R$ and $N$ don't overlap in a way that forces the operator to "mix" them. Once you establish that independence of $R$ and $N$ guarantees such a decomposition exists, the second part asks you to show that $N$ itself is forced to be that unique complement, which follows from the rank-nullity relationship and the invariance properties already in play.

The core idea is understanding when a subspace can be "split off" cleanly by the operator itself. A $T$-invariant complement to $R$ means we can decompose $V = R \oplus W$ where $T$ maps $W$ into itself—this is possible precisely when $R$ and $N$ don't overlap in a way that forces the operator to "mix" them. Once you establish that independence of $R$ and $N$ guarantees such a decomposition exists, the second part asks you to show that $N$ itself is forced to be that unique complement, which follows from the rank-nullity relationship and the invariance properties already in play.