Example of a Non-Subspace in R^2 Closed Under Addition and Inverses?

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Discussion Overview

The discussion revolves around identifying a subset of R² that is closed under vector addition and taking additive inverses but is not a subspace of R². Participants explore the definitions and properties of vector spaces and subspaces in the context of this problem.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the problem and seeks help in finding an example.
  • Another participant suggests considering subsets of R² that are subspaces, implying a need for clarity on the definitions involved.
  • Some participants note that while addition is possible, the issue may relate to the properties of the subset that are not fully aligned with the definition of a vector space.
  • A participant mentions the importance of closure under addition and additive inverses, indicating that the subset must include the zero vector.
  • Another participant proposes a trivial subspace, {(0,0)}, but admits to struggling with the original problem.
  • One participant suggests solving a simpler problem regarding nontrivial subgroups of the group of real numbers, hinting at a potential method to find the desired subset.
  • It is noted that for a subset to be a subspace, it must also be closed under scalar multiplication, suggesting that this aspect may be relevant to the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific example of a subset that meets the criteria. Multiple viewpoints and interpretations of the problem are presented, indicating ongoing uncertainty and exploration.

Contextual Notes

There are unresolved aspects regarding the definitions of vector spaces and the specific properties required for a subset to qualify as a subspace. The discussion includes assumptions about closure under addition and inverses without fully resolving the implications of these properties.

Who May Find This Useful

This discussion may be of interest to students and educators in mathematics, particularly those studying linear algebra and vector spaces.

mrroboto
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I don't understand this, can someone help?:

What is an example of a subset of R^2 which is closed under vector addition and taking additive inverses which is not a subspace of R^2?


R, in this question, is the real numbers.

Thanks!
 
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Can you think of a subset of R^2 such that it is a subspace?
 
Well you can add things but you can't subtract things. This should be a big enough hint.
 
ZioX said:
Well you can add things but you can't subtract things.

He mentioned additive inverses and closure under addition (implying 0 being an element), so he has no problem subtracting things. The key lies in the one part of the definition of a vector space that is left out.
 
slider142 said:
He mentioned additive inverses and closure under addition (implying 0 being an element), so he has no problem subtracting things. The key lies in the one part of the definition of a vector space that is left out.

I misread it. I thought the question asked to find something that fails to be a subspace because it's not closed under additive inverses.
 
To JasonRox: Yes, I can think of a subset, V, of R^2 that is a subspace. V= {(0,0)}. But I still don't understand how to approach this problem.
 
first try to solve a simpler problem.
are there nontrivial subgroups of the group [tex](\mathbb{R},+,-,0)[/tex] ? that means is there a set [tex]A[/tex] with [tex]\{0\} \subset A \subset \mathbb{R}[/tex] such that [tex]A[/tex] is closed under addition and substraction?
once you have found such a subgroup [tex]A[/tex], is [tex]A^2[/tex] a set with the desired property?
 
In order that a subset be a subspace, it must be closed under addition, additive inverses, and scalar multiplication. Since your subset is required to be closed under addition and additive inverse, there's only one place left to look!
 

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