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Could a saint please guide me work through my research on AFFINE SETS AND MAPPINGS 
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#1
Feb2612, 07:11 AM

P: 74

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 nonempty 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 nonempty subset M of V with the property that λx+(1λ)y ε M whenever x,y ε M and λ ε ℝ To illustrate this concept, show that: M = { x = (x_{1},...x_{4}) ε ℝ^{4} : 2x_{1}x_{2}+x_{3} = 1 and x_{1}+4x_{3}2x_{4} = 3} is an affine subset of ℝ^{4}. I'm not so sure where to start. Opinions welcome Regards Tam 


#2
Feb2612, 08:05 AM

Mentor
P: 18,345

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_1z_2+z_3=1~\text{and}~z_1+4z_22z_4=3[/tex] You know that [itex]z_i=\lambda x_i + (1\lambda) y_i[/itex] so substitute that in. 


#3
Feb2712, 09:50 AM

P: 74

So subbing in z we get: LHS: 2(λx_{1}+(1λ)y_{1})  (λx_{2}+(1λ)y_{2}) + (λx_{3}+(1λ)y_{3}) Now taking: λ(2x_{1}x_{2}+x_{3}) We know that the bold part = 1 (1λ)(2y_{1}y_{2}+y_{3}) We again know that the bold part = 1 so we have λ + (1λ) = 1 = RHS AND NOW DO THE SAME WITH THE SECOND PART X1+4X32X4 = 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 


#4
Feb2712, 10:23 AM

P: 74

Could a saint please guide me work through my research on AFFINE SETS AND MAPPINGS
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 = m_{1}+a and y = m_{2}+a for some m_{1},m_{2}M Therefore, λ(m_{1}+a) + (1λ)(m_{2}+a) Now rearranging gives: (i) λm_{1} + (1λ)m_{2} which must be in M by definition. (ii) λa + (1λ)a =a(λ+1λ) =a Hence, λm_{1} + (1λ)m_{2} + a is in M+a. So M+a is affine. I'm unsure of what to do with the zero part of the question? 


#5
Feb2712, 11:39 AM

Mentor
P: 18,345

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.



#6
Feb2812, 09:13 AM

P: 74

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 


#7
Feb2812, 09:17 AM

Mentor
P: 18,345

Begin by showing (iii). Apply the definition of M affine on x and 0. 


#8
Feb2812, 09:33 AM

P: 74

λx + (1λ)x will be in M by definition on 0, λ(0) + (1λ)(0) = 0 which is in M since we are assuming 0 ε M Have I understood you? 


#9
Feb2812, 09:37 AM

Mentor
P: 18,345

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+(1λ)y is in M. Now choose a special value of y. 


#10
Feb2912, 09:36 AM

P: 74

Then we get (lambda)x + (1lambda)(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 


#11
Feb2912, 09:38 AM

Mentor
P: 18,345

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). 


#12
Feb2912, 09:53 AM

P: 74

Or could we use the M+a proof? 


#13
Mar112, 04:39 AM

P: 74

bump?



#14
Mar112, 04:43 AM

Sci Advisor
P: 906

is it permissible to set λ = 1/2?



#15
Mar112, 07:25 AM

P: 74

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 


#16
Mar212, 02:56 PM

Sci Advisor
P: 906

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..... 


#17
Mar412, 08:54 AM

P: 74

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 


#18
Mar812, 10:48 AM

P: 1

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 


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