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Questions › DSE › DSE 2021 › Q8
4 marks, −1Mathematics › Vector spaces, rank and eigenvalues › Linear independence, span and basisHardabout 3 min

DSE 2021 Question 8: Vector spaces, rank and eigenvalues

Consider the vector space Rn\mathbb{R}^nRn for some n≥2n \geq 2n≥2, over the field of real numbers, and the following two statements. Statement A: {v1,v2,…,vn−1,vn}\{v_1, v_2, \dots, v_{n-1}, v_n\}{v1​,v2​,…,vn−1​,vn​} are linearly independent vectors. Statement B: {(v1+v2),(v2+v3),…,(vn−1+vn),(vn+v1)}\{(v_1 + v_2), (v_2 + v_3), \dots, (v_{n-1} + v_n), (v_n + v_1)\}{(v1​+v2​),(v2​+v3​),…,(vn−1​+vn​),(vn​+v1​)} are linearly independent vectors. Which of the following statements is true?

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Plain-text version

Consider the vector space ℝⁿ for some n ≥ 2, over the field of real numbers, and the following two statements. Statement A: {v₁, v₂, …, vₙ₋₁, vₙ} are linearly independent vectors. Statement B: {(v₁ + v₂), (v₂ + v₃), …, (vₙ₋₁ + vₙ), (vₙ + v₁)} are linearly independent vectors. Which of the following statements is true? (A) A implies B and B implies A (B) A implies B if n is odd, and B always implies A (C) A implies B if n is even, and B always implies A (D) A always implies B and B implies A if n is even

Also written as: Consider the vector space R^n for some n >= 2, over the field of real numbers, and the following two statements. Statement A: {v_1, v_2,..., v_(n-1), v_n} are linearly independent vectors. Statement B: {(v_1 + v_2), (v_2 + v_3),..., (v_(n-1) + v_n), (v_n + v_1)} are linearly independent vectors. Which of the following statements is true?

Vector spaces, rank and eigenvalues has come up 41 times in 6 exams since 2008.

What this question tests

This question is on Linear independence, span and basis, part of Vector spaces, rank and eigenvalues in Mathematics.

Whether vectors depend on each other, what they span and when they form a basis.

TestStop has 13 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2021 · Q20Let W be the subspace of ℝ³ spanned by (0, -1, 2) and (2, -1, 0). Which one of the following vectors lies in…
  • IIT JAM 2022 · Q6Let W be a subspace of a vector space ℝ³. Then which of the following sets of vectors forms a basis of W?
Practise Vector spaces, rank and eigenvaluesMore on Linear independence, span and basis

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