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Math Help - Are this an affine subset of R^n?

  1. #1
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    Are this an affine subset of R^n?


    L=\{x= \[\alpha_1,\ldots,\alpha_n\]^\top\  | \sum_{i=1}^n \alpha_i^2 =615\}

    And in general how can I show wether a subset of a vector space is affine or not?

    Thank you!
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  2. #2
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    Re: Are this an affine subset of R^n?

    A subset S, of vector space V, is an "affine" set if and only if for \vec{v} a member of S, the set \{\vec{u}- \vec{v}|u\in X\} is a subspace of V.
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  3. #3
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    Re: Are this an affine subset of R^n?

    What is your X here?
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  4. #4
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    Re: Are this an affine subset of R^n?

    Whatever the overall space is. In your problem, it would be R^n. As long as you are working in R^n you can charactize an "affine set" (that is not a subspace), geometrically, as a line or plane or "hyper-plane" (of whatever dimension) that does NOT include the origin. The condition that \sum \alpha_i^2= 615 simply says that this is the set of all vectors in R^n with length \sqrt{615}. Is that a plane (of whatever dimension) in R^n?
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  5. #5
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    Re: Are this an affine subset of R^n?

    Hmm, yes I tihnk that is a plane. By the way I found an other way of solution but Im not sure wether it is right or not. So L \subseteq \mathbb{R}^n is affine if \alpha x+\beta y \in L for all x,y\in L and  \alpha + \beta=1.

    And what I did:
     \sum_{i=1}^n \alpha {x_i}^2 + \sum_{i=1}^n (1-\alpha) {y_i}^2
    then follows
    \alpha \sum_{i=1}^n  {x_i}^2 + (1-\alpha) \sum_{i=1}^n  {y_i}^2
    and because \sum_{i=1}^n {x_1}^2 =615 and \sum_{i=1}^n {y_1}^2=615

    \alpha \sum_{i=1}^n  {x_i}^2 + (1-\alpha) \sum_{i=1}^n  {y_i}^2 =615
    So it is an affine set.
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  6. #6
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    Re: Are this an affine subset of R^n?

    You should have: \sum_{i = 1} (\alpha x_i + (1-\alpha) y_i)^2
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