The GATE Grind

GATE 2025 DA – Question 25

Linear Algebra · Vector spaces, linear independence, rank and nullity · 1 mark · Multiple select

Which of the following statements is/are correct?

  1. $\mathbb{R}^n$ has a unique set of orthonormal basis vectors
  2. $\mathbb{R}^n$ does not have a unique set of orthonormal basis vectors
  3. Linearly independent vectors in $\mathbb{R}^n$ are orthonormal
  4. Orthonormal vectors $\mathbb{R}^n$ are linearly independent

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Correct answer: (B) $\mathbb{R}^n$ does not have a unique set of orthonormal basis vectors; (D) Orthonormal vectors $\mathbb{R}^n$ are linearly independent

Explanation

Rotating or reflecting an orthonormal basis gives another orthonormal basis, so there is no unique one (B is true, A is false). Linearly independent vectors need not be of unit length or perpendicular, such as $(1, 0)$ and $(1, 1)$, so C is false. Orthonormal vectors are linearly independent: in $\sum c_iv_i = 0$ taking the inner product with $v_j$ gives $c_j = 0$ for every $j$ (D is true).