= Solution
For the three given <qutrit> <vectors>, $\langle\phi_0|=(\langle0|-i\langle2|)/\sqrt2$ and $\langle\phi_2|=(\langle0|+i\langle2|)/\sqrt2$. Taking the <complex conjugate> in the bra is essential. Their squared <norms> are $1,1,1$, and
$$
\langle\phi_0|\phi_2\rangle=\tfrac12\{1+(-i)(-i)\}=0,\qquad \langle\phi_0|\phi_1\rangle=\langle\phi_2|\phi_1\rangle=0.
$$
Thus 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 \b[it is an <orthonormal basis>].
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