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

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

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

    belongs to$\displaystyle M$.

    ii) A sum of the form (1) is called an affine combination of $\displaystyle x_{1}, x_{2}, ... x_{n}$. Prove that, given a non-empty subset $\displaystyle S$ of $\displaystyle V$, the set consisting of all afine combinations of elements of $\displaystyle S$ is an affine subset of $\displaystyle V$ and is the smallest affine subset of $\displaystyle V$ containing $\displaystyle S$. This set is called the affine span of $\displaystyle S$.
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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 $\displaystyle V$ is a non-empty subset $\displaystyle M$ of $\displaystyle V$ with the property that $\displaystyle \lambda x+(1-\lambda )y \in M$ whenever $\displaystyle x, y \in M$ and $\displaystyle \lambda \in \Re$

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

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

    belongs to$\displaystyle M$.

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

    1. The base case "works" quite easily.

    2. Assuming the result holds for $\displaystyle n-1$, as $\displaystyle \sum_{i=1}^{n} \lambda_{i} = 1$ you want to set $\displaystyle \lambda = \lambda_n$ for your induction step. You then have to show that the rest of the sum is of the form $\displaystyle (1 - \lambda)y$ with $\displaystyle y \in M$. 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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