Chapter 9.3, Problem 43E

To determine

(a)

To show:

The equality that $\left(x,y\right)=x^T\overline{y}$ where $\overline{y}=\text{col}\left(\overline{y_1},\overline{y_2},\cdots \overline{y_n}\right)$.

To determine

(b)

To prove:

For any $n\times 1$ vectors $x$, $y$, $z$ and any complex number $\lambda$, we have

$\left(x,y\right)=\overline{\left(y,x\right)}$

$\left(x,y+z\right)=\left(x,y\right)+\left(x,z\right)$.

$\left(\lambda x,y\right)=\lambda \left(x,y\right)$, $\left(x,\lambda y\right)=\overline{\lambda }\left(x,y\right)$.