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Could a saint please guide me work through my research on AFFINE SETS AND MAPPINGS

  1. Feb 26, 2012 #1
    This research project is to help me (I'm an undergraduate) get my head around this topic. It is concerned with affine subsets of a vector space and the mappings between them. As an
    application, the construction of certain fractal sets in the plane is considered. It would be considered pretty basic to a seasoned maths student.

    I am wanting to learn this so I will be sticking around. I will not just leave. I want to commit to this. Thanks

    There are two parts: A and B

    If someone is willing to help, I will post each topic AFTER I have fully understood the previous topic. This way it will run in a logical order.

    PART A:

    ----------------------------------------------------------------------------------
    Throughout Part A, V will be a real vector space and, for a non-empty subset S of V and
    a ε V , the set {x+a: x ε S} will be denoted by S + a

    ----------------------------------------------------------------------------------

    TOPIC 1: Definition of Affine Subset:

    An affine subset of V is a non-empty subset M of V with the property that λx+(1-λ)y ε M whenever x,y ε M and λ ε ℝ

    To illustrate this concept, show that:

    M = { x = (x1,...x4) ε ℝ4 : 2x1-x2+x3 = 1 and x1+4x3-2x4 = 3}

    is an affine subset of ℝ4.

    I'm not so sure where to start. Opinions welcome

    Regards
    Tam
     
  2. jcsd
  3. Feb 26, 2012 #2

    micromass

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Take x and y in M. You must show that [itex]\lambda x+ (1-\lambda) y\in M[/itex]. Call this number z for convenience.

    To show that z is in M, you need to show that

    [tex]2z_1-z_2+z_3=1~\text{and}~z_1+4z_2-2z_4=3[/tex]

    You know that [itex]z_i=\lambda x_i + (1-\lambda) y_i[/itex] so substitute that in.
     
  4. Feb 27, 2012 #3
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Okay great.

    So subbing in z we get:

    LHS:

    2(λx1+(1-λ)y1) - (λx2+(1-λ)y2) + (λx3+(1-λ)y3)

    Now taking:

    λ(2x1-x2+x3) We know that the bold part = 1

    (1-λ)(2y1-y2+y3) We again know that the bold part = 1


    so we have λ + (1-λ) = 1 = RHS

    AND NOW DO THE SAME WITH THE SECOND PART X1+4X3-2X4 = 3... IVE DONE THAT IN MY OWN TIME.
    -------------------------------------------------------------------------------------------------------

    So I think I've grasped that. I will look at the next topic and report back when Ive had a go. Thanks Micro
     
  5. Feb 27, 2012 #4
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Now Topic A2

    Let M be an affine subset of V.

    QUESTION: Prove that M+a is affine for every a ε V and that, if 0 ε M, then M is a subspace

    So my attempt:

    Proof: x,y is in M+a

    take: x = m1+a and y = m2+a for some m1,m2M

    Therefore, λ(m1+a) + (1-λ)(m2+a)

    Now rearranging gives:

    (i) λm1 + (1-λ)m2 which must be in M by definition.

    (ii) λa + (1-λ)a
    =a(λ+1-λ)
    =a

    Hence, λm1 + (1-λ)m2 + a is in M+a. So M+a is affine.


    I'm unsure of what to do with the zero part of the question?
     
  6. Feb 27, 2012 #5

    micromass

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    So assume that 0 is in M. You must prove that it is a subspace. So you must check the axioms of being a subspace.
     
  7. Feb 28, 2012 #6
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    OKay, using the definition: Let M be a subspace of vecotr space V. Then M is a subspace of V IFF

    i) 0 ε M
    ii) x+y ε M for all x,y ε M
    iii) λx ε M for all x ε M

    (i) holds since we are assuming 0 ε M
    (ii) holds since we showed this in the last part of the question
    (iii) holds since in the last part of the question λx ε M

    Is this enough? I'm unsure of (iii)

    Regards
    Tam
     
  8. Feb 28, 2012 #7

    micromass

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Could you explain (ii) and (iii)?? You have to use the assumption that 0 is in M for all of these questions.
    Begin by showing (iii). Apply the definition of M affine on x and 0.
     
  9. Feb 28, 2012 #8
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Okay, on x,

    λx + (1-λ)x will be in M by definition

    on 0,

    λ(0) + (1-λ)(0) = 0 which is in M since we are assuming 0 ε M

    Have I understood you?
     
  10. Feb 28, 2012 #9

    micromass

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    No. You have to show that for any x and for any λ, that λx is in M.

    You know that M is affine, so you know that for any x and for any y, we have that λx+(1-λ)y is in M.
    Now choose a special value of y.
     
  11. Feb 29, 2012 #10
    Oh okay. If we take y=0 (using the condition 0€M)

    Then we get (lambda)x + (1-lambda)(0) which is just (lambda)x

    So we know for any x and lambda that it will be in M. So that is iii done.

    What about ii

    Ps: I'm on my phone so sorry for weak notation.

    Thanks micro
     
  12. Feb 29, 2012 #11

    micromass

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    For (ii), you need to prove that if x and y are in M, then x+y is in M.

    You know that for each r and s in M that

    [tex]\lambda r+(1-\lambda)s\in M[/tex]

    Now choose the right r and s such that we can conclude that x+y is in M. Use (iii).
     
  13. Feb 29, 2012 #12
    Take r=x and s=0 so since we know (lambda)x is in M, x+0 is in M.

    Or could we use the M+a proof?
     
  14. Mar 1, 2012 #13
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    bump?
     
  15. Mar 1, 2012 #14

    Deveno

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    is it permissible to set λ = 1/2?
     
  16. Mar 1, 2012 #15
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Yes, I think...

    Edit: taking λ = 1/2

    f(x+y) = f(1/2(2x)) + f(1/2(2y))

    = 1/2 [ f(2x) + f(2y) ]

    taking 2 out gives:

    = f(x) + f(y)

    Hence closed under addition

    Is that sufficient?

    Thanks
     
    Last edited: Mar 1, 2012
  17. Mar 2, 2012 #16

    Deveno

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    where does "f" come from?

    my reasoning goes like this: 1/2 and 1/2 sum to 1, so (1/2)x + (1/2)y is an affine combination, that is: (x+y)/2 is in M.

    now, use part (iii) to conclude that.....
     
  18. Mar 4, 2012 #17
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Yeah forget about the 'f's.. yeah that makes sense.

    Deveno, if I sent you the question sheet it may be easier for both you and I to understand. Of course, only if you are happy to help. Would that be okay? The reason I ask is that it is hard for me to get my points across since I don't know latex.

    Regards
     
  19. Mar 8, 2012 #18
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    Hi

    I am doing a similar assignment and have been finding it difficult to find relevant material to the questions. However I have found the guidance on this thread very useful so far and was hoping you could send me any further information on this assignment as I think it would be a great help.

    Thanks
     
  20. Mar 8, 2012 #19
    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

    I think this is the same assignment as the one I'm doing - I'm basically in the same position as matt90, and have spent hours doing research on this with no luck. I would also really appreciate any additional help you have to offer.
     
  21. Mar 9, 2012 #20

    berkeman

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    Re: Could a saint please guide me work through my research on AFFINE SETS AND MAPPING

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