GATE DA 2025 | Question 25

MSQ · 1 mark

Which of the following statements is/are correct?

  1. Option A: \(\mathbb{R}^{n}\) has a unique set of orthonormal basis vectors
  2. Option B: \(\mathbb{R}^{n}\) does not have a unique set of orthonormal basis vectors (correct)
  3. Option C: Linearly independent vectors in \(\mathbb{R}^{n}\) are orthonormal
  4. Option D: Orthonormal vectors \(\mathbb{R}^{n}\) are linearly independent (correct)

Answer

(B), (D)

Explanation

  • A and B: orthonormal bases are not unique. Rotating the standard basis gives another one; in \(\mathbb{R}^{2}\), \(\frac{1}{\sqrt{2}}(1, 1)\) and \(\frac{1}{\sqrt{2}}(1, -1)\) also form an orthonormal basis. Even changing the sign of one vector gives a new basis. So A is false and B is true.
  • C is false: \((1, 0)\) and \((1, 1)\) are linearly independent, but they are not orthogonal and \((1, 1)\) is not a unit vector.
  • D is true: if \(\sum_{i} c_{i}v_{i} = 0\) for orthonormal \(v_{i}\), taking the dot product with \(v_{j}\) leaves \(c_{j} = 0\), so every coefficient is zero.

Answer: B, D