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Math Help - Basis linear algebra question

  1. #1
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    Basis linear algebra question

    I have a really long homework question that I need help with. There are a couple of parts to it so I appreciate anyone that can help out.

    It starts by just stating that a basis v_1,\cdots,v_n \in \mathbb{R}^n is an orthonormal basis if the following holds for all i, j = 1,\cdots, n:

    v_i \cdot v_j = \begin{cases} 0 & i \neq j \\ 1 & i=j \end{cases}

    Assuming this, I have to:

    a) Show that for any x \in \mathbb{R}^n we have:

    x=(v_1 \cdot x)v_1+ \cdots +(v_n \cdot x)v_n

    b) Show that:

    v_1 = \frac{1}{sqrt{2}} (1, 0, 1)

    v_2 = \frac{1}{sqrt{3}} (1, 1, -1)

    v_3 = \frac{1}{sqrt{6}} (-1, 2, 1)

    is an orthonormal basis in [tex]\mathbb{R}^3[\tex] and expand the vector x=(10, -12, 3) in the basis v_1, v_2, v_3.

    c) Show that for any x \in \mathbb{R}^n:

    x \cdot x = (v_1 \cdot x)^2 + \cdots + (v_n \cdot x)^2

    and for any x, y \in \mathbb{R}^n:

    x \cdot y = (v_1 \cdot x)(v_1 \cdot y) + \cdots + (v_n \cdot x)(v_n \cdot y)





    As I said, this seems like a lot to go through and appreciate any help since I literally am stuck!
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: Basis linear algebra question

    Your images don't link right. Can you post them again?

    -Dan
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  3. #3
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    Re: Basis linear algebra question

    Quote Originally Posted by topsquark View Post
    Your images don't link right. Can you post them again?

    -Dan
    Trying to fix them now, the tex commands appear correct though? Only some tex commands appear.

    EDIT: hmmm all the tex commands with spaces just do not work, do you have an idea?
    Last edited by MichaelH; November 27th 2013 at 05:52 PM.
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