Question

Why isn't the (complex) dot product of 2 vectors (x and y) defined as sum_{k=0}^{n}(\Re(x_{k})*\Re(y_{k})+\Im(x_{k})*\Im(y_{k})) ?

Answer 1

\(\sum_{k=0}^{n}(\Re(x_{k})*\Re(y_{k})+\Im(x_{k})*\Im(y_{k})) \)

What does that mean?

Answer 2

\[%x_k = \Re(x_k)+\Im(x_k),\ \qquad y_k = \Re(y_k)+\Im(y_k)\\[2ex]
x_k\cdot y_k=\Big(\Re(x_k)+\Im(x_k)\Big)\cdot\Big(\Re(y_k)+\Im(y_k)\Big)\\
\qquad\quad = \Re(x_k)\cdot\Re(y_k)+\Re(x_k)\cdot\Im(y_k)+\Im(x_k)\cdot\Re(y_k)+\Im(x_k)\cdot\Im(y_k)\]