The complex inner product is conjugate symmetric, not anticommutative: $\langle x,y\rangle=\overline{\langle y,x\rangle}$. With $\langle x,y\rangle=x^*y$, conjugating $\sum_j\overline{y_j}x_j$ gives the required expression. Anticommutativity would introduce a minus sign and is a different property.
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