Linear Algebra Done Right · Problem 7F.14

The key insight is that the operator norm $\|P_U - P_W\|$ measures how "far apart" two projection operators are geometrically. When this distance is strictly less than 1, the projections are similar enough that they must preserve dimension. Think about what happens when you project onto $U$ versus $W$: if the difference in these projections is small, vectors can't be "lost" or "gained" in the transition, so the subspaces must have the same dimensionality. The proof hinges on showing that $\|P_U - P_W\| < 1$ prevents either subspace from being strictly smaller than the other.

The key insight is that the operator norm $\|P_U - P_W\|$ measures how "far apart" two projection operators are geometrically. When this distance is strictly less than 1, the projections are similar enough that they must preserve dimension. Think about what happens when you project onto $U$ versus $W$: if the difference in these projections is small, vectors can't be "lost" or "gained" in the transition, so the subspaces must have the same dimensionality. The proof hinges on showing that $\|P_U - P_W\| < 1$ prevents either subspace from being strictly smaller than the other.