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GATE DA 2025 Q25: Orthonormal bases and linear independence in ℝⁿ

Question

Which of the following statements is/are correct?

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Answer: B, D (official GATE 2025 final answer key)

Concept used: Orthonormal bases are not unique; orthonormality implies linear independence, not the reverse.

First step: Rotate the standard basis of R2\mathbb{R}^2 by any angle: you get another orthonormal basis. For independence, take the dot product of a zero linear combination with each orthonormal vector.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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