Inner Product

Printable View

December 8th 2010, 08:56 AM
alice8675309

Inner Product

For the inner product spaces V (over F) and linear transformations g: V->F, find a vector y such that g(x)=<x,y> for all x in V.

(1) V=R^3, g(a_1,a_2,a_3)=a_1-2a_2+4a_3

(2) V=P2(R) with <f,h>= integral 0 to1 f(t)h(t)dt, g(f)=f(0)+f'(1)

ps: i'm trying my best with the latex, but for some reason when i use it the page gets all distorted.
December 8th 2010, 10:05 AM
tonio

Quote:

Originally Posted by alice8675309

For the inner product spaces V (over F) and linear transformations g: V->F, find a vector y such that g(x)=<x,y> for all x in V.

(1) V=R^3, g(a_1,a_2,a_3)=a_1-2a_2+4a_3

(2) V=P2(R) with <f,h>= integral 0 to1 f(t)h(t)dt, g(f)=f(0)+f'(1)

ps: i'm trying my best with the latex, but for some reason when i use it the page gets all distorted.

Suppose $y:=(y_1,y_2,y_3)\,,\,and\,\,\forall x=(x_1,x_2,x_3)\in\mathbb{R}^3\,,\,\,g(x):=x_1-2x_2+4x_3=x_1y_1+x_2y_2+x_3y_3=\langle x,y\rangle$ ,

then choosing $x_2=x_3=0$ , we get $y_1=1$ , choosing $x_1=x_3=0\,,\,\,y_2=-2$ , and choosing

$x_1=x_2=0\,,\,\,y_3=4$ , so $y=(1,-2,4)$ .

Now you do (2) by yourself.

Tonio
December 18th 2010, 10:30 AM
alice8675309

Quote:

Originally Posted by tonio

Suppose $y:=(y_1,y_2,y_3)\,,\,and\,\,\forall x=(x_1,x_2,x_3)\in\mathbb{R}^3\,,\,\,g(x):=x_1-2x_2+4x_3=x_1y_1+x_2y_2+x_3y_3=\langle x,y\rangle$ ,

then choosing $x_2=x_3=0$ , we get $y_1=1$ , choosing $x_1=x_3=0\,,\,\,y_2=-2$ , and choosing

$x_1=x_2=0\,,\,\,y_3=4$ , so $y=(1,-2,4)$ .

Now you do (2) by yourself.

Tonio

Thanks! But for "umber 2 I went and found an orthonormal basis to the inner product using gram schmidt and got {1, 2sqt(3)(x-1/2), 6sqt(5)(x^2-x+1/6}. Then I went on like number one and I'm stuck. I'm not coming anywhere close to the answer given in the back of the book. I hope someone can give me some direction.
Just in case, the answer given is: 210x^2-204x+33
December 18th 2010, 01:15 PM
Opalg

Quote:

Originally Posted by alice8675309

Thanks! But for "umber 2 I went and found an orthonormal basis to the inner product using gram schmidt and got {1, 2sqt(3)(x-1/2), 6sqt(5)(x^2-x+1/6}. Then I went on like number one and I'm stuck. I'm not coming anywhere close to the answer given in the back of the book. I hope someone can give me some direction.
Just in case, the answer given is: 210x^2-204x+33

For no. 2, don't use Gram–Schmidt. Do it like this.

Let $f(x) = ax^2+bx+c$ . Then $f(0) + f'(1) = 2a+b+c$ . So we want to find $h(x) = px^2+qx+r\in P_2(\mathbb{R})$ such that

$\displaystyle \langle f,h\rangle = \int_0^1(ax^2+bx+c)(px^2+qx+r)\,dx = 2a+b+c.$

Work out that integral, then find the values of p, q and r that give the right result $2a+b+c$ for it.