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Math Help - Inner Product

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by alice8675309 View Post
    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
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    Quote Originally Posted by tonio View Post
    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
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  4. #4
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    Quote Originally Posted by alice8675309 View Post
    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.
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