GATE DA 2025 Set 1 — Question 25
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Linear Algebra → Vector Algebra → Basis & Dimension
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Question
Which of the following statements is/are correct?
Correct answer
(B) Rⁿ does not have a unique set of orthonormal basis vectors; (D) Orthonormal vectors Rⁿ are linearly independent
Solution
Statement (A) is incorrect because any rotation of an orthonormal basis in results in another orthonormal basis. Thus, the set is not unique.
Statement (B) is correct for the reason above.
Statement (C) is incorrect because linearly independent vectors are not necessarily orthogonal or of unit length (e.g., and in ).
Statement (D) is correct because orthogonality implies linear independence for non-zero vectors.
Statement (B) is correct for the reason above.
Statement (C) is incorrect because linearly independent vectors are not necessarily orthogonal or of unit length (e.g., and in ).
Statement (D) is correct because orthogonality implies linear independence for non-zero vectors.
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