GATE 2025 DA Q.25: solution and answer

GATE 2025 · DAQ.25 · 1 mark · MSQ

Which of the following statements is/are correct?

(A)
$R^{n}$ has a unique set of orthonormal basis vectors
(B)
$R^{n}$ does not have a unique set of orthonormal basis vectors
(C)
Linearly independent vectors in $R^{n}$ are orthonormal
(D)
Orthonormal vectors $R^{n}$ are linearly independent

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Answer and solution

ANSWER

The correct statements are (B) and (D).

Official answer: B, DSource: Official answer key, GATE 2025 DA (IIT Roorkee)

Test (A) and (B). In $R^{2}$ , ${(1, 0), (0, 1)}$ is an orthonormal basis. So is ${\frac{1}{2} (1, 1), \frac{1}{2} (- 1, 1)}$ : each vector has length $\frac{1}{2} 2 = 1$ and their dot product is $\frac{1}{2} (- 1 + 1) = 0$ . In general, if ${e_{1}, \dots, e_{n}}$ is orthonormal and $Q$ is any orthogonal matrix, then ${Q e_{1}, \dots, Q e_{n}}$ is another orthonormal basis, because $(Q e_{i})^{⊤} (Q e_{j}) = e_{i}^{⊤} Q^{⊤} Q e_{j} = δ_{ij}$ . So the orthonormal basis is not unique: (A) is false and (B) is true.
Test (C). In $R^{2}$ the vectors $(1, 0)$ and $(1, 1)$ are linearly independent, since neither is a multiple of the other. But $(1, 0) \cdot (1, 1) = 1 \neq = 0$ , so they are not orthogonal, and hence not orthonormal. (C) is false.
Test (D). Let $v_{1}, \dots, v_{k}$ be orthonormal, so $v_{j}^{⊤} v_{i} = δ_{ij}$ . Suppose $\sum_{i} c_{i} v_{i} = 0$ . Multiply on the left by $v_{j}^{⊤}$ to get $\sum_{i} c_{i} v_{j}^{⊤} v_{i} = c_{j} = v_{j}^{⊤} \cdot 0 = 0$ . This holds for every $j$ , so all coefficients are zero. Orthonormal vectors are linearly independent: (D) is true.
Correct statements: (B) and (D).

KEY IDEA

Orthonormal vectors always satisfy $v_{j}^{⊤} v_{i} = δ_{ij}$ , so taking the inner product of any zero combination with $v_{j}$ forces $c_{j} = 0$ . This gives linear independence. The converse is false, and an orthonormal basis can be rotated by any orthogonal matrix, so it is never unique.

WATCH OUT FOR

Treating linear independence and orthonormality as equivalent (picking C), or assuming the standard basis is the only orthonormal basis of $R^{n}$ (picking A).

From the GATE 2025 DA paper. Every question of this paper · Download the full paper · All 3 Vector spaces questions, with notes