In the textbook that I'm using the standard inner product is defined as
$$\langle x,y\rangle = \sum_{i=1}^{n}a_{i}\overline{b_{i}}$$
where $x=(a_{1}, a_{2}, {...}, a_{n})$ and $y=(b_{1}, b_{2}, {...}, b_{n})$ and the bar denotes complex conjugation.
In one the practice problems I was doing they used the fact that
$\langle x,y\rangle = y^{*}x$
where $y^{*}$ denotes the conjugate transpose of y. I'm having trouble understanding why the definition for the standard inner product is the same as $y^{*}x$.
I tried doing an example where $x = (i,i)$ and $y = (-i,-i)$
For $$\langle x,y\rangle = i*i+i*i=-2$$
On the other hand
$y^{*}x=\begin{pmatrix} i\\i\\ \end{pmatrix} \begin{pmatrix} i & i\\ \end{pmatrix} = \begin{pmatrix} -1 & -1\\ -1 & -1\\ \end{pmatrix} $
I was wondering why I am getting $\langle x,y\rangle \ne y^{*}x$