2 marks, −0.667MathematicsVector spaces, rank and eigenvaluesLinear independence, span and basisModerateabout 2 min

IIT JAM 2021 Question 20: Vector spaces, rank and eigenvalues

Let $W$ be the subspace of $\mathbb{R}^3$ spanned by $(0, -1, 2)$ and $(2, -1, 0)$ . Which one of the following vectors lies in $W$ ?

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Answer and worked solution

Answer: (C)

A general element of $W$ is $a(0,-1,2) + b(2,-1,0) = (2b,\ -a-b,\ 2a)$ .

For $(1,-1,1)$ : $2b = 1$ and $2a = 1$ give $a = b = \tfrac12$ , and then $-a - b = -1$ , matching the middle entry. So $(1,-1,1)$ lies in $W$ , option (C). The other candidates fail the middle coordinate.

The trap: (A)

(1, 1, 1)

Matching only the first and third coordinates is not enough; the middle entry $-a-b$ must also fit, which rules out (A), (B) and (D).

Plain-text version

Let W be the subspace of ℝ³ spanned by (0, -1, 2) and (2, -1, 0). Which one of the following vectors lies in W? (A) (1, 1, 1) (B) (-1, 1, 1) (C) (1, -1, 1) (D) (1, 1, -1)

Also written as: Let W be the subspace of R^3 spanned by (0, -1, 2) and (2, -1, 0). Which one of the following vectors lies in W?

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 2022 · Q6Let W be a subspace of a vector space ℝ³. Then which of the following sets of vectors forms a basis of W?

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