Understand Affine Subsets & Mappings: Research Project for Undergrads

In summary: Basically, if you have a subset of a vector space, and you want to show that the subset is affine, you need to show that the following three things hold: First, the subset has a nonzero element; second, adding an element to the subset doesn't change its size; and third, adding an element to the subset changes its shape.
  • #1
tamintl
74
0
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
 
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  • #2


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.
 
  • #3


micromass said:
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.

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 I've had a go. Thanks Micro
 
  • #4


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?
 
  • #5


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


micromass said:
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.

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
 
  • #7


tamintl said:
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

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.
 
  • #8


micromass said:
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.

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?
 
  • #9


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.
 
  • #10
micromass said:
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.

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
 
  • #11


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
micromass said:
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).

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?
 
  • #13


bump?
 
  • #14


is it permissible to set λ = 1/2?
 
  • #15


Deveno said:
is it permissible to set λ = 1/2?

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:
  • #16


tamintl said:
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

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...
 
  • #17


Deveno said:
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...

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
 
  • #18


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
 
  • #19


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.
 
  • #20


Thread closed temporarily for Moderation...
 

1. What is an affine subset?

An affine subset is a subset of a vector space that is closed under affine combinations. In simpler terms, an affine subset is a set of points that can be formed through linear combinations of a fixed set of points in a vector space.

2. How is an affine subset different from a subspace?

An affine subset differs from a subspace in that it does not necessarily contain the origin of the vector space. This means that an affine subset is not closed under scalar multiplication, unlike a subspace.

3. What is an affine mapping?

An affine mapping, also known as an affine transformation, is a type of function that preserves the properties of affine subsets. It maps an affine subset in one vector space to another affine subset in a different vector space, while preserving the relationships between the points in the original set.

4. How are affine subsets and mappings used in research projects for undergraduates?

Affine subsets and mappings are commonly used in research projects for undergraduates, particularly in the fields of mathematics, computer science, and engineering. They provide a framework for studying and analyzing geometric objects and their transformations.

5. What are some real-world applications of affine subsets and mappings?

Affine subsets and mappings have numerous real-world applications, such as computer graphics, robotics, and computer vision. They are also used in economics and finance for analyzing market trends and forecasting. Additionally, affine transformations are used in physics and engineering for modeling and analyzing physical systems.

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