http://i37.tinypic.com/118j0p1.jpg
uppon what law this transition was made?
how did they deside what is the real and what is the imaginate part?
http://i37.tinypic.com/118j0p1.jpg
uppon what law this transition was made?
how did they deside what is the real and what is the imaginate part?
The "decide" what are the real and imaginary parts of a number from the definition: If x= a+ bi then the real part of x is a and the imaginary part is b.
They are also using the "linearity" of the inner product,
<au+ bv, w>= a<u, w>+ b<v, w>. and the fact that $\displaystyle <u, v>= \overline{<u, v>}$ so that $\displaystyle <u, av+ bw>= \overline<av+ bw, u>= \overline{a<v,u>}+ \overline{b<w,u>}$$\displaystyle = \overline{a}<u, v>+ \overline{b}<u, w>$.