Final answer:
The statement "V=V" is trivially true as any set is always equal to itself. However, it lacks meaningful information about the relationship between subspaces U and V. To establish equality, their bases should be identical, which the statement doesn't guarantee. Consequently, it doesn't imply that U and V are equal.
Step-by-step explanation:
The statement "If U and V are subspaces of ℝ^n, and B is a basis for U and V, then V=V" is tautologically true but lacks informative content.
The assertion V=V is trivially accurate because any set is always equal to itself.
However, it doesn't convey any meaningful information about the relationship between U and V. For U and V to be equal, their bases should be identical, not just denoted by the same symbol B.
The statement does not provide evidence that the subspaces U and V share the same basis, and thus, it cannot be concluded that V equals U.
In summary, the statement is trivially true but does not offer insights into the relationship between the subspaces U and V based on the given information.
3. True or false For each part, determine if you believe the statement is true or false, and justify your answers as appropriate. (a) If \( U \) and \( V \) are subspaces of \( \mathbb{R}^{n} \), then V=V.