Question

Which of the following statements is/are correct?

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

Hint:

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.

Answer 1

The Correct Option is B, D

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).