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Math Help - [SOLVED] Inner product question

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    MHF Contributor arbolis's Avatar
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    [SOLVED] Inner product question

    I've just learned we can define an inner product "almost" as we want. I mean by that that it must satisfy the four conditions to be an inner product. My question is : How many different inner products can we define? Is there an infinity of them, or are there a finite number? If there is a finite number of them, how many are them?
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    Hi,

    Quote Originally Posted by arbolis View Post
    I've just learned we can define an inner product "almost" as we want. I mean by that that it must satisfy the four conditions to be an inner product. My question is : How many different inner products can we define? Is there an infinity of them, or are there a finite number? If there is a finite number of them, how many are them?
    I can't prove it, but I'd say there is an infinity of them...

    http://www.math.jussieu.fr/%7Enekovar/co/q/quad.pdf << page 31, exercise 2.3.9 at the bottom of the page. And in class, we encountered several inner products, based on quadratic forms with various coefficients...
    (you can find a French version of this in the same site )
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    Quote Originally Posted by arbolis View Post
    I've just learned we can define an inner product "almost" as we want. I mean by that that it must satisfy the four conditions to be an inner product. My question is : How many different inner products can we define? Is there an infinity of them, or are there a finite number? If there is a finite number of them, how many are them?
    Well, if you consider the Euclidean space \mathbb{R}^n, we can define an inner product between two vectors \textbf{u} = \left(u_1,\;u_2,\;\dots,\;u_n\right) and \textbf{v} = \left(v_1,\;v_2,\;\dots,\;v_n\right) as

    \langle\textbf{u},\;\textbf{v}\rangle = a_1u_1v_1 + a_2u_2v_2 + a_3u_3v_3 + \cdots + a_nu_nv_n for any a_i>0\in\mathbb{R} (try checking this).

    That already gives us an infinite amount (by varying our choices of the a_is), and there are also infinitely many other vector spaces for which we can define inner products.
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    MHF Contributor arbolis's Avatar
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    Thanks to both. I also have another question : the elements of a vector space are called "vectors", right? Also, a matrix, a polynomial can be elements of a vector space. This implies that a matrix and polynomials can be called "vectors"?
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    Quote Originally Posted by arbolis View Post
    Thanks to both. I also have another question : the elements of a vector space are called "vectors", right? Also, a matrix, a polynomial can be elements of a vector space. This implies that a matrix and polynomials can be called "vectors"?
    Yes. By definition, a vector is an element of a vector space, so a matrix or a polynomial can indeed be considered a vector. However, outside of the context of general vector spaces, "vector" usually refers to a vector in Euclidean n-space, so be clear what you mean when you refer to vectors in this way.
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