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Math Help - proof by induction of affine subsets

  1. #1
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    proof by induction of affine subsets

    Hi, I've got a project at uni and part of it is this proof by induction which i am terrible at. I can never work out where to sub in n+1 or n+2 n-1 or whatever it may be so any help much appreciated!!

    An affine subset of V is a non-empty subset M of V with the property that \lambda x+(1-\lambda )y \in M whenever x, y \in M and \lambda \in \Re

    i) Let M be an affine subset of V. Prove by induction on n that, if x_{1}, x_{2}, ... x_{n} \in M and \lambda_{1}, \lambda_{2}, ... \lambda_{n} \in \Re with \sum_{i=1}^{n} \lambda_{i} = 1, then

    x=\sum_{i=1}^{n} \lambda_{i}x_{i} (1)

    belongs to M.

    ii) A sum of the form (1) is called an affine combination of x_{1}, x_{2}, ... x_{n}. Prove that, given a non-empty subset S of V, the set consisting of all afine combinations of elements of S is an affine subset of V and is the smallest affine subset of V containing S. This set is called the affine span of S.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by leshields View Post
    Hi, I've got a project at uni and part of it is this proof by induction which i am terrible at. I can never work out where to sub in n+1 or n+2 n-1 or whatever it may be so any help much appreciated!!

    An affine subset of V is a non-empty subset M of V with the property that \lambda x+(1-\lambda )y \in M whenever x, y \in M and \lambda \in \Re

    i) Let M be an affine subset of V. Prove by induction on n that, if x_{1}, x_{2}, ... x_{n} \in M and \lambda_{1}, \lambda_{2}, ... \lambda_{n} \in \Re with \sum_{i=1}^{n} \lambda_{i} = 1, then

    x=\sum_{i=1}^{n} \lambda_{i}x_{i} (1)

    belongs to M.

    ii) A sum of the form (1) is called an affine combination of x_{1}, x_{2}, ... x_{n}. Prove that, given a non-empty subset S of V, the set consisting of all afine combinations of elements of S is an affine subset of V and is the smallest affine subset of V containing S. This set is called the affine span of S.
    Two hints for the first part:

    1. The base case "works" quite easily.

    2. Assuming the result holds for n-1, as \sum_{i=1}^{n} \lambda_{i} = 1 you want to set \lambda = \lambda_n for your induction step. You then have to show that the rest of the sum is of the form (1 - \lambda)y with y \in M. This, I believe, requires a bit of a trick, but think about it for a bit first.

    Hopefully that should help you with the induction bit though...
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