Orthogonal Projection Operator and a Subset

Question

I'm looking for help on answering this question, relatively straightforward for most of you but new to me:

Let H₁ be the orthogonal projection operator onto S₁, and let H₀ be the orthogonal projection operator onto S₀ ⊆ S₁. Explain why H₁ − H₀ is the orthogonal projection operator onto the orthogonal complement of S₀ within S₁.

I think the answer is related to the idea that the projector onto the plane W perpendicular to V - for example - is the identity minus the projector onto the span of V:

$$P_W=\mathbb{I}-P_V$$

Therefore, since my question is dealing with the space S₁ and not the whole space then logically if follows that H₁ − H₀ is the orthogonal projection operator onto the orthogonal complement of S₀ within S₁

Let me know if anyone agrees with this step.

Answer 1

Your intuition is correct.

To precise it, just notice that the restriction to $S_1$ of the orthogonal projection onto $S_1$ is the identity.