Discussion Overview
The discussion centers on the subspace W of C[0,1] spanned by the set S={sin^2(x), cos^2(x), cos2x}. Participants explore the linear independence of the functions in S and seek to identify a basis for W.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Homework-related
Main Points Raised
- One participant claims that S is not a basis for W because it is not linearly independent.
- Another suggests using trigonometric identities to express one function in terms of the others to identify which function is superfluous.
- A later reply emphasizes the need to show which functions are superfluous to establish a basis.
- Another participant notes that since the functions span a space of fewer than three dimensions, the dimension must be 0, 1, or 2, and concludes that 0 or 1 is nonsensical, implying that two functions must be independent.
Areas of Agreement / Disagreement
Participants generally agree that the functions in S are not linearly independent and that a basis for W can be found, but there is no consensus on the specific basis or the method to derive it.
Contextual Notes
The discussion does not resolve the specific mathematical steps needed to identify the basis or clarify the dependencies among the functions in S.