Is this complex vector orthogonal to itself?

[Steve Turchin]

Is the basis vector $(i,0,1)$ in the space $V=$Span$((i,0,1))$ with a standard inner product,over $\mathbb{C}^3$
orthogonal to itself?
$<(i,0,1),(i,0,1)> = i \cdot i + 0 \cdot 0 + 1 \cdot 1 = -1 + 1 = 0$
The inner product (namely dot product) of this vector with itself is equal to zero.
What is going on here?

 

[Orodruin]

No, it is not orthogonal to itself. The inner product on a complex vector space must be anti-linear in one of the arguments (which one depends on whether you use physics or maths notation). In other words, the complex inner product is given by
$$
\langle x, y \rangle = \sum_k x_k^* y_k.
$$