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Questions › DSE › DSE 2011 › Q3
1 markMathematics › Vector spaces, rank and eigenvalues › Linear independence, span and basisModerateabout 2 min

DSE 2011 Question 3: Vector spaces, rank and eigenvalues

Suppose {v1,v2,…,vn}\{v_{1},v_{2},\ldots,v_{n}\}{v1​,v2​,…,vn​} is a set of linearly dependent vectors, none of them being the zero vector. Suppose c1,c2,…,cnc_{1},c_{2},\ldots,c_{n}c1​,c2​,…,cn​ are scalars, not all zero, such that ∑i=1ncivi=0\sum_{i=1}^{n}c_{i}v_{i}=0∑i=1n​ci​vi​=0. Then the minimum number of non-zero scalars is

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Suppose {v₁,v₂,…,vₙ} is a set of linearly dependent vectors, none of them being the zero vector. Suppose c₁,c₂,…,cₙ are scalars, not all zero, such that Σᵢ₌₁ⁿcᵢvᵢ=0. Then the minimum number of non-zero scalars is (A) 1 (B) 2 (C) n-1 (D) cannot be determined.

Also written as: Suppose {v_1,v_2,...,v_n} is a set of linearly dependent vectors, none of them being the zero vector. Suppose c_1,c_2,...,c_n are scalars, not all zero, such that sum_(i=1)^nc_iv_i=0. Then the minimum number of non-zero scalars is

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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Question from the DSE 2011 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.