# inner product not by standart basis

• Jul 22nd 2011, 08:04 AM
transgalactic
inner product not by standart basis
we choose a basis B={v1=(1,0)v2=(1,1)} from $\displaystyle R^2$
and define $\displaystyle (x,y)_B=x_1y_1+x_2y_2$

why does the prof say that
$\displaystyle x=x_1v_1+x_2v_2$
$\displaystyle y=y_1v_1+y_2v_2$
how did he get those formulas?
and i cant se the structure of our chosen basis here?
how to see it in thos formulas
?
• Jul 22nd 2011, 10:16 AM
FernandoRevilla
Re: inner product not by standart basis
Quote:

Originally Posted by transgalactic
we choose a basis B={v1=(1,0)v2=(1,1)} from $\displaystyle R^2$ and define $\displaystyle (x,y)_B=x_1y_1+x_2y_2$ why does the prof say that $\displaystyle x=x_1v_1+x_2v_2$ $\displaystyle y=y_1v_1+y_2v_2$

Although I don't know what is the final question, taking into account that $\displaystyle B$ is a basis of $\displaystyle \mathbb{R}^2$ , every $\displaystyle x\in \mathbb{R}^2$ can be uniquely expressed in the way $\displaystyle x=x_1v_1+x_2v_2$ . Same considerations for $\displaystyle y$ .
• Jul 22nd 2011, 10:29 AM
transgalactic
Re: inner product not by standart basis
if x=(x1,x2) then x could be represented as $\displaystyle x=x_1v_1+x_2v_2$

only if v1=(1,0) v2=(0,1)

but here we have a different basis not a standart one.
so our coefficient is not x1 x2

??
• Jul 22nd 2011, 12:21 PM
FernandoRevilla
Re: inner product not by standart basis
Perhaps this simple example will help you. Consider $\displaystyle x=(5,2)$ and $\displaystyle y=(2,1)$ . Prove that $\displaystyle x=3v_1+2v_2$ and $\displaystyle y=1v_1+1v_2$ . According to the given definition of the inner product we have $\displaystyle (x|y)=3\cdot 1+2\cdot 1=5$ .
• Jul 22nd 2011, 01:04 PM
transgalactic
Re: inner product not by standart basis
ok so if i will go by your example
(x,y)_B=x1y1+x2y2=5*2+2*1=12

x=(5,2)=5(1,0)+2(1,1)=(7,2)

i try to follow the logic but i cant understand this term
• Jul 22nd 2011, 11:09 PM
FernandoRevilla
Re: inner product not by standart basis
Quote:

Originally Posted by transgalactic
x=(5,2)=5(1,0)+2(1,1)=(7,2)

$\displaystyle (5,2)\neq (7,2)$
• Jul 22nd 2011, 11:41 PM
transgalactic
Re: inner product not by standart basis
i know
i got here some thing wrong
just cant understand this basis of inner product
i am looking for a tutorial with basic examples