GATE 2020 EC Q.1: solution and answer

GATE 2020 · ECQ.1 · 1 mark · MCQ

If $v_{1}, v_{2}, \dots, v_{6}$ are six vectors in $R^{4}$ , which one of the following statements is FALSE?

(A)
It is not necessary that these vectors span $R^{4}$ .
(B)
These vectors are not linearly independent.
(C)
Any four of these vectors form a basis for $R^{4}$ .
(D)
If ${v_{1}, v_{3}, v_{5}, v_{6}}$ spans $R^{4}$ , then it forms a basis for $R^{4}$ .

See the question in the paper

Answer and solution

ANSWER

The answer is (C). Four vectors chosen from six in $R^{4}$ can be linearly dependent, so "any four of them form a basis" does not hold in general. (A), (B) and (D) are all true.

Official answer: CSource: Official answer key, GATE 2020 EC (IIT Delhi)

Key fact: $dim R^{4} = 4$ . So a linearly independent set in $R^{4}$ has at most 4 vectors, and a spanning set has at least 4 vectors.
(A) Six vectors need not span $R^{4}$ . For example, if all six are multiples of $e_{1}$ , their span is only a line. So (A) is TRUE.
(B) Six vectors in a 4-dimensional space are always linearly dependent, because $6 > 4$ . Equivalently, the $4 \times 6$ matrix $A = [v_{1} \dots v_{6}]$ has rank at most $4 < 6$ , so $A x = 0$ has a nontrivial solution. So (B) is TRUE.
(C) (C) says that every choice of four vectors is a basis, so one counterexample disproves it. Take $v_{1} = e_{1}$ and $v_{2} = 2 e_{1}$ , with $v_{3}, \dots, v_{6}$ anything. The set ${v_{1}, v_{2}, v_{3}, v_{4}}$ satisfies $2 v_{1} - v_{2} = 0$ , so it is linearly dependent and cannot be a basis. So (C) is FALSE.
(D) In an $n$ -dimensional space, a set of exactly $n$ vectors that spans the space is automatically linearly independent, and so it is a basis. Here ${v_{1}, v_{3}, v_{5}, v_{6}}$ has 4 vectors and spans $R^{4}$ , so it is a basis. So (D) is TRUE.
The only FALSE statement is (C).

KEY IDEA

In an $n$ -dimensional space: more than $n$ vectors are always dependent, a spanning set needs at least $n$ vectors, and a set of exactly $n$ vectors is a basis as soon as it spans or is independent (either one implies the other). Having more than $n$ vectors says nothing about whether a particular subset of $n$ of them is independent.

WATCH OUT FOR

Confusing "some four of the vectors form a basis" (possible, if the six span $R^{4}$ ) with "any four form a basis" (false in general), and so picking (C) as true. A second trap is reading (A) as "the vectors do not span $R^{4}$ ". It only says spanning is not guaranteed, which is true.

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GATE 2020 EC Q.1: solution and answer

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