Gate DA 2025 | Question - 25 | Linear Algebra

(A) $\mathbb{R}^n$ has a unique set of orthonormal basis vectors
❌ Incorrect
There are infinitely many orthonormal bases in $\mathbb{R}^n$. The Gram-Schmidt process can be applied to many different linearly independent sets to produce different orthonormal bases.

(B) $\mathbb{R}^n$ does not have a unique set of orthonormal basis vectors
✅ Correct
As stated above, many orthonormal bases exist. So this statement is true.

(C) Linearly independent vectors in $\mathbb{R}^n$ are orthonormal
❌ Incorrect
Linear independence does not imply orthonormality. For example, vectors $[1, 0]$ and $[1, 1]$ in $\mathbb{R}^2$ are linearly independent but not orthonormal.

(D) Orthonormal vectors in $\mathbb{R}^n$ are linearly independent
✅ Correct
Any set of orthonormal vectors is automatically linearly independent by definition.