List of top Linear Algebra Questions on Inner Product Spaces and Orthonormal Basis asked in GATE MA

Let \(P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 16 \end{pmatrix}\). Define an inner product on \(\mathbb{R}^3\) with respect to \(P\) as \[ \langle x,y\rangle_P = x^{\top}Py, \quad \text{for all } x,y \in \mathbb{R}^3. \] Consider the subspace \(V = \text{span}\left\{\begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}1\\1\\1\end{pmatrix}\right\}\). Which one of the following sets is an orthonormal basis of \(V\) with respect to \(\langle x,y\rangle_P\)?