I am playing with complex vectors and inner products but I am stuck on a very simple example where something is clearly wrong, but I cannot find out where exactly I am committing a mistake.
I am considering the space of all 2-dimensional complex column vectors, with the inner product \begin{equation*} \left\langle r,s\right\rangle =s^{\ast }r \end{equation*} where $s^{\ast }$ is the conjugate transpose of $s$.
The set of two vectors \begin{equation*} b_{1}=\left[ \begin{array}{c} \frac{1}{\sqrt{2}} \\ \frac{i}{\sqrt{2}} \end{array} \right] \text{ }b_{2}=\left[ \begin{array}{c} \frac{i}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \end{array} \right] \end{equation*} is orthonormal, that is, the two vectors are orthogonal and have unit norm (you can easily see it, and I double checked in Matlab).
The vectors of an orthonormal set should be linearly independent. However, in this case, I get \begin{equation*} b_{1}+ib_{2}=0 \end{equation*}
In other words, the two vectors are linearly dependent, as the above linear combination yields the zero vector, but the coefficients of the linear combination are not all zero.
Where is the mistake?