Discussion Overview
The discussion revolves around the question of whether two orthogonal subsets \( S_{1} \) and \( S_{2} \) of \( \mathbb{R}^{n} \) that span the same subspace \( W \) can be shown to be equal by demonstrating mutual inclusion. Participants explore methods for proving the equality of spans of these subsets.
Discussion Character
- Technical explanation
- Mathematical reasoning
Main Points Raised
- One participant suggests that showing \( S_{1} \subset S_{2} \) and \( S_{2} \subset S_{1} \) would suffice to prove \( S_{1} = S_{2} \) and thus that they span the same space.
- Another participant counters that while this method works if the sets are equal, it is unlikely to be the case and proposes instead to show \( S_{1} \subset \text{span}(S_{2}) \), which implies \( \text{span}(S_{1}) \subset \text{span}(S_{2}) \).
- A later reply acknowledges a correction in the notation used and reiterates the importance of showing the spans are equal.
- Another participant points out that since the sets are orthogonal, knowing that one span is contained in the other is sufficient due to the dimension being equal to the number of elements in the sets.
- One participant expresses understanding and appreciation for the clarification provided.
Areas of Agreement / Disagreement
Participants generally agree that showing mutual inclusion is not necessary if the spans are equal and that the orthogonality of the sets simplifies the proof. However, there is no consensus on the best method to demonstrate the equality of spans, with multiple approaches being discussed.
Contextual Notes
Some assumptions about the properties of orthogonal sets and their spans are discussed, but the implications of these assumptions on the methods proposed remain unresolved.