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Questions › DSE › DSE 2011 › Q4
1 markMathematics › Vector spaces, rank and eigenvalues › Linear independence, span and basisModerateabout 2 min

DSE 2011 Question 4: Vector spaces, rank and eigenvalues

Suppose the non-zero n×1n\times 1n×1 column vector xxx solves the system of equations Ax=bAx=bAx=b, where AAA is an m×nm\times nm×n matrix whose columns are the vectors a1,a2,…,ana_{1},a_{2},\ldots,a_{n}a1​,a2​,…,an​, and bbb is an m×1m\times 1m×1 column vector. Then the set of vectors {a1,a2,…an,b}\{a_{1},a_{2},\ldots a_{n},b\}{a1​,a2​,…an​,b} is

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Suppose the non-zero n× 1 column vector x solves the system of equations Ax=b, where A is an m× n matrix whose columns are the vectors a₁,a₂,…,aₙ, and b is an m× 1 column vector. Then the set of vectors {a₁,a₂,… aₙ,b} is (A) linearly independent. (B) linearly dependent. (C) linearly dependent only if a₁,a₂,…,aₙ are linearly dependent. (D) linearly dependent only if m=n.

Also written as: Suppose the non-zero n* 1 column vector x solves the system of equations Ax=b, where A is an m* n matrix whose columns are the vectors a_1,a_2,...,a_n, and b is an m* 1 column vector. Then the set of vectors {a_1,a_2,... a_n,b} 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.