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Questions › DSE › DSE 2008 › Q60
2 marksMathematics › Vector spaces, rank and eigenvalues › Linear independence, span and basisHardabout 4 min

DSE 2008 Question 60: Vector spaces, rank and eigenvalues

Suppose v1,v2v_{1},v_{2}v1​,v2​ and v3v_{3}v3​ are three vectors in 3-dimensional space and are linearly dependent. Then the vectors v1+v2, v2+v3v_{1}+v_{2},\ v_{2}+v_{3}v1​+v2​, v2​+v3​, and v1+v3v_{1}+v_{3}v1​+v3​

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Suppose v₁,v₂ and v₃ are three vectors in 3-dimensional space and are linearly dependent. Then the vectors v₁+v₂, v₂+v₃, and v₁+v₃ (A) are linearly independent (B) may be linearly dependent (C) are linearly dependent (D) are linearly independent, except when one of the vectors is zero.

Also written as: Suppose v_1,v_2 and v_3 are three vectors in 3-dimensional space and are linearly dependent. Then the vectors v_1+v_2, v_2+v_3, and v_1+v_3

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 2008 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.