1. ## Inner Product Space

Could someone explain to me how to go from (1) to (2)?

$\displaystyle ||x - \langle x,y\rangle y||^2 = \langle x - \langle x,y\rangle y,x - \langle x,y\rangle y\rangle$

$\displaystyle = \langle x,x\rangle - \langle x,\langle x,y\rangle y\rangle - \langle \langle x,y\rangle y,x\rangle + \langle \langle x,y\rangle y,\langle x,y\rangle y\rangle$ (1)

$\displaystyle ||x||^2 - \overline{\langle x,y\rangle}\langle x,y\rangle - \langle x,y\rangle\langle y,x\rangle + \langle x,y\rangle\overline{\langle x,y\rangle}$ (2)

$\displaystyle ||x||^2 - |\langle x,y\rangle|^2$.

(Just before this it states assume $\displaystyle ||y|| = 1$)

2. Hello there!

Could someone explain to me how to go from (1) to (2)?

$\displaystyle ||x - \langle x,y\rangle y||^2 = \langle x - \langle x,y\rangle y,x - \langle x,y\rangle y\rangle$

$\displaystyle = \langle x,x\rangle - \langle x,\langle x,y\rangle y\rangle - \langle \langle x,y\rangle y,x\rangle + \langle \langle x,y\rangle y,\langle x,y\rangle y\rangle$ (1)

$\displaystyle ||x||^2 - \overline{\langle x,y\rangle}\langle x,y\rangle - \langle x,y\rangle\langle y,x\rangle + \langle x,y\rangle\overline{\langle x,y\rangle}$ (2)

$\displaystyle ||x||^2 - |\langle x,y\rangle|^2$.

(Just before this it states assume $\displaystyle ||y|| = 1$)
There are some rules by definition of the dot product,

like <x,x> = ||x||^2

By definition it is

$\displaystyle <x, \lambda y> = \overline{\lambda}<x,y>$, but

$\displaystyle <\lambda x, y> = \lambda<x,y>,$

So let $\displaystyle \lambda := <x,y>$

$\displaystyle <x, <x,y>y > = \overline{<x,y>}<x,y>$

And that is why $\displaystyle <<x,y>y,x > = <x,y><y,x>$

And the last one

$\displaystyle <<x,y>y,<x,y>y> = \overline{<x,y>}<x,y><y,y>$

$\displaystyle = \overline{<x,y>}<x,y>||y||^2$

and $\displaystyle ||y||^2 = 1 = ||y||$ so

$\displaystyle = \overline{<x,y>}<x,y>$

Yours
Rapha