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Questions › DSE › DSE 2015 › Q9
1 mark, −0.333333333333333Mathematics › Vector spaces, rank and eigenvalues › Linear independence, span and basisHardabout 4 min

DSE 2015 Question 9: Vector spaces, rank and eigenvalues

Vectors v0,v1,…,vnv_0,v_1,\dots,v_nv0​,v1​,…,vn​ in Rm\mathbb{R}^mRm are affinely independent if ∑i=0ncivi=0\sum_{i=0}^n c_iv_i=0∑i=0n​ci​vi​=0 and ∑i=0nci=0\sum_{i=0}^n c_i=0∑i=0n​ci​=0 imply ci=0c_i=0ci​=0 for all iii. For such a set, which is implied? I. v0,…,vnv_0,\dots,v_nv0​,…,vn​ are linearly independent. II. (v1−v0),…,(vn−v0)(v_1-v_0),\dots,(v_n-v_0)(v1​−v0​),…,(vn​−v0​) are linearly independent. III. n≤mn\le mn≤m.

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Vectors v₀,v₁,…,vₙ in ℝ^m are affinely independent if Σᵢ₌₀ⁿ cᵢvᵢ=0 and Σᵢ₌₀ⁿ cᵢ=0 imply cᵢ=0 for all i. For such a set, which is implied? I. v₀,…,vₙ are linearly independent. II. (v₁-v₀),…,(vₙ-v₀) are linearly independent. III. n≤ m. (A) Only I and II are true (B) Only I and III are true (C) Only II is true (D) Only II and III are true

Also written as: Vectors v_0,v_1,...,v_n in R^m are affinely independent if sum_(i=0)^n c_iv_i=0 and sum_(i=0)^n c_i=0 imply c_i=0 for all i. For such a set, which is implied? I. v_0,...,v_n are linearly independent. II. (v_1-v_0),...,(v_n-v_0) are linearly independent. III. n<= m.

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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