Discussion Overview
The discussion revolves around the question of proving that the set of non-invertible matrices is unbounded. Participants explore definitions and properties related to non-invertible matrices, particularly focusing on the determinant and norms of matrices.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Homework-related
Main Points Raised
- One participant attempts to relate the determinant of a matrix to the concept of unboundedness by suggesting that the line y=0 (representing det(A)=0) is unbounded.
- Another participant questions the initial claim by stating that for the analogy to hold, the set of non-invertible matrices must be a subset of real numbers.
- A challenge is posed to find a non-invertible matrix with a norm greater than 10, prompting a discussion about the choice of norms.
- Several participants note that the determinant is not a norm and argue that if it were, the set of non-invertible matrices would actually be bounded.
- There is an acknowledgment that there are infinitely many norms for matrices, and participants suggest checking specific course materials for the appropriate norm to use.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the unboundedness of the set of non-invertible matrices, with some arguing for boundedness based on the properties of the determinant, while others challenge this perspective and emphasize the need for clarity on norms.
Contextual Notes
There is uncertainty regarding the definition of norms for matrices and the implications of using the determinant in this context. The discussion highlights the need for clarity on mathematical definitions and assumptions.