Proving vector spaces where (a1a2 < equal to 0)

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Homework Help Overview

The discussion revolves around the properties of a vector set denoted as , where the condition (a1a2 ≤ 0) is specified. Participants are exploring whether this set can be considered a vector space, particularly focusing on closure under scalar multiplication and vector addition.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants are questioning the definitions and properties of the set, including whether represents ordered pairs in ℝ2. There are attempts to clarify the implications of the condition (a1a2 ≤ 0) on vector addition and scalar multiplication. Some express uncertainty about the meaning of terms used, such as "negative vector," and the context of the vector space in question.

Discussion Status

The discussion is active, with participants offering hints and asking clarifying questions. There is a recognition of the need for more precise definitions and understanding of the set's properties. Some participants suggest exploring counterexamples and writing down the sum of two arbitrary members of the subset to further investigate the problem.

Contextual Notes

There is ambiguity regarding the notation used for the vector set and the specific vector space being referenced. Participants are grappling with the implications of the condition (a1a2 ≤ 0) and its effect on the axioms of vector spaces.

maiad
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Homework Statement


For the vector set<a1,a2>, where (a1a2 < equal to 0)


Homework Equations





The Attempt at a Solution



I'm not sure why this set is close under scalar multiplication and not in vector addition. Some hints would be nice :D
 
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Is <a1,a2> your notation for ordered pairs? In other words, are you trying to define a subset of ℝ2? In that case, it should be easy to find a counterexample.

I don't think it's possible to give you a hint without completely solving the problem for you.
 
Well i think for vector addition, it's open because there's no negative vector of a1 or a2 since a1a1<0? am I correct? and I'm assuming a1 is a vector, and a2 is another vector.
 
Then I don't understand the definition of the set. What vector space is this supposed to be a subset of? If a1 and a2 are vectors, what does <a1,a2> and a1a2 mean? I also don't understand the sentence "there's no negative vector of a1 or a2".
 
um the set is a notation for ordered pairs, I was trying to refer to the axiom for vector addition that states" For each x in V, there exist a vector -x such that x+(-x)=(-x)+x=0" is not satisfied
 
You said that a1 and a2 are vectors. In what vector space? Or would you like to change that and say that they are real numbers instead? Because if they are vectors, I don't know what a1a2 means.
 
Why don't you start by just writing down the sum of two arbitrary members of this subset?
 

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