QS-020 · Linear Algebra · Easy
GATE DA 2025 | Question: 15
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 in $\mathbb{R}^n$ are linearly independent.
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 in $\mathbb{R}^n$ are linearly independent.
Solution
Statement 1: False
An orthonormal basis of $\mathbb{R}^n$ is not unique. For example, $\mathbb{R}^2$ has infinitely many orthonormal bases obtained by rotation.
Statement 2: True
Hence, $\mathbb{R}^n$ does not have a unique orthonormal basis.
Statement 3: False
Linear independence does not imply orthogonality or unit length.
Statement 4: True
Every set of mutually orthogonal nonzero vectors is linearly independent. Orthonormal vectors are therefore linearly independent.
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