Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 1 / a / Solution

For the three given qutrit vectors, $⟨{\varphi }_0|=(⟨0|-i⟨2|)/\sqrt{2}$ and $⟨{\varphi }_2|=(⟨0|+i⟨2|)/\sqrt{2}$. Taking the complex conjugate in the bra is essential. Their squared norms are $1,1,1$, andThus they are an orthonormal set of three vectors in the three-dimensional inner product space ${\mathbb{C}}^3$. Such a set is linearly independent and spans the space, so it is an orthonormal basis.