Question:
Which of the following statements is/are correct?
Which of the following statements is/are correct?
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Orthonormal vectors are guaranteed to be linearly independent, and they form a basis for the space. If vectors are orthonormal, they are automatically linearly independent.
Updated On: Feb 15, 2026
- \( \mathbb{R}^n \) has a unique set of orthonormal basis vectors
- \( \mathbb{R}^n \) does not have a unique set of orthonormal basis vectors
- Linearly independent vectors in \( \mathbb{R}^n \) are orthonormal
- Orthonormal vectors in \( \mathbb{R}^n \) are linearly independent
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The Correct Option is B, D
Solution and Explanation
Let's evaluate each option:
Option (A): \( \mathbb{R}^n \) does not have a unique set of orthonormal basis vectors. There can be different sets of orthonormal basis vectors, which are related by rotation or reflection. Thus, this statement is incorrect.
Option (B): This is correct. \( \mathbb{R}^n \) does not have a unique set of orthonormal basis vectors because any orthonormal set of vectors can be transformed by an orthogonal matrix into another orthonormal set. Therefore, this statement is true.
Option (C): Linearly independent vectors in \( \mathbb{R}^n \) are not necessarily orthonormal. For a set of vectors to be orthonormal, they must be not only linearly independent but also have unit length and be mutually orthogonal. Hence, this option is incorrect.
Option (D): Orthonormal vectors in \( \mathbb{R}^n \) are always linearly independent because they have unit length and are orthogonal to each other. Therefore, this statement is correct.
Thus, the correct answer is (B) and (D).
Option (A): \( \mathbb{R}^n \) does not have a unique set of orthonormal basis vectors. There can be different sets of orthonormal basis vectors, which are related by rotation or reflection. Thus, this statement is incorrect.
Option (B): This is correct. \( \mathbb{R}^n \) does not have a unique set of orthonormal basis vectors because any orthonormal set of vectors can be transformed by an orthogonal matrix into another orthonormal set. Therefore, this statement is true.
Option (C): Linearly independent vectors in \( \mathbb{R}^n \) are not necessarily orthonormal. For a set of vectors to be orthonormal, they must be not only linearly independent but also have unit length and be mutually orthogonal. Hence, this option is incorrect.
Option (D): Orthonormal vectors in \( \mathbb{R}^n \) are always linearly independent because they have unit length and are orthogonal to each other. Therefore, this statement is correct.
Thus, the correct answer is (B) and (D).
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