Discussion
Let W be the subspace of \(\mathbf{R}^{4}\) given by :
\[ W = \left\{ \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{bmatrix} \middle/ \begin{aligned} &x_{1} + 2x_{2} + 3x_{3} + 4x_{4} = 0 \\ &x_{1} - x_{2} + x_{3} = 0 \end{aligned} \right\} \]
Then \(W^{\perp}\) is generated by :
Answer: B
\(W^{\perp}\) is generated by \((1, 2, 3, 4)^{t}\) and \((1, -1, 1, 0)^{t}\).
W is the set of vectors orthogonal to \(a = (1, 2, 3, 4)\) and \(b = (1, -1, 1, 0)\), since its two equations say \(a \cdot x = 0\) and \(b \cdot x = 0\). So \(W = \mathrm{span}\{a, b\}^{\perp}\), and in a finite dimensional space \(W^{\perp} = \mathrm{span}\{a, b\}\). The two vectors are independent, so \(\dim W^{\perp} = 2\) and \(\dim W = 2\).
Option A spans a 3-dimensional space, too large. Options C and D give vectors that are not in the span of \(a\) and \(b\); for instance \((0, 0, 0, 1)\) would need a combination of \(a, b\) with zero first three entries, which forces both coefficients to be 0.
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