Discussion
Let V be a finite dimensional vector space over K and W be a subspace of V. Then which of the following statements is true ?
Answer: C
The true statement is \(\dim V = \dim W + \dim V/W\).
The quotient map \(V \to V/W\) is onto with kernel W, so by rank-nullity \(\dim V = \dim W + \dim V/W\). Equivalently, extend a basis of W to a basis of V; the added vectors give a basis of V/W.
W/V is not defined, since W is the smaller space. A minus sign, or leaving out \(\dim W\), gives the wrong count; for \(V = \mathbf{R}^{3}\) and W a plane, \(\dim V/W = 1\) and only \(3 = 2 + 1\) works.
No comment is present. Be the first to comment.
Loading…
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