Discussion
Let \(f : \mathbf{R}^{2} \to \mathbf{R}^{3}\) be a linear map and
\(A = \{(x, y) \in \mathbf{R}^{2} / |x| \le 1, |y| \le 1\}\).
Then :
Answer: A
\(f(A)\) is a connected subset of R3.
A is the closed square \([-1, 1] \times [-1, 1]\), which is connected (indeed convex) and compact. A linear map is continuous, and the continuous image of a connected set is connected. In fact \(f(A)\) is convex, since a linear map sends line segments to line segments.
The other options fail. \(f(A)\) is the continuous image of a compact set, so it is compact and in particular bounded, not unbounded. The image of R2 under a linear map into R3 lies in a subspace of dimension at most 2, a closed set with empty interior, so \(f(A)\) can be neither open in R3 nor dense in it.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development