Difference between a vector subspace and subset?

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Homework Help Overview

The discussion revolves around the concepts of vector subspaces and subsets within the context of vector spaces, particularly focusing on the properties that define a vector subspace. Participants are exploring the distinctions between these two concepts, especially in relation to vectors that point into the first quadrant of a 2D vector space.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to clarify the definition of a vector subspace versus a subset, with specific examples involving vectors in the first quadrant. Questions arise regarding whether certain subsets can be considered vector subspaces based on their closure under vector operations.

Discussion Status

The discussion is ongoing, with participants questioning the properties of subsets of vectors and their compliance with the requirements of vector spaces. Some have offered insights into the closure properties of these subsets, while others express confusion about the implications of their interpretations.

Contextual Notes

There is a focus on the specific example of vectors pointing into the first quadrant, with participants examining the implications of this choice on the classification of these vectors as subspaces. The discussion reflects a lack of consensus on the definitions and properties being debated.

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This is quite confusing to me. I know a vector subspace is a vector space within another vector space and is closed under the operations of the vector space it lies in, but how exactly does it differ from vector subsets? Anyone care to explain or clarify this? My textbook is completely terrible at explaining this. Thanks in advance.
 
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Consider the subset of all vectors that point into the first quadrant.
Is that a vector subspace?
 
robphy said:
Consider the subset of all vectors that point into the first quadrant.
Is that a vector subspace?

So, in this case, the vector space V would be the entire first quadrant. Since the vectors that point into the first quadrant would imply they have components not in the first quadrant, that would mean that it is not a vector subspace, correct?
 
Take a 2D vector space (a plane).
The subset of all vectors [with base point at the origin] that point into the first quadrant have positive x- and y-components. Do those vectors satisfy the properties of a vector space?
 
Retribution said:
So, in this case, the vector [STRIKE]space[/STRIKE] subset V would be the entire first quadrant. Since the vectors that point into the first quadrant would imply they have components not in the first quadrant, that would mean that it is not a vector subspace, correct?
Not sure what that means.
 
robphy said:
Take a 2D vector space (a plane).
The subset of all vectors [with base point at the origin] that point into the first quadrant have positive x- and y-components. Do those vectors satisfy the properties of a vector space?
It's closed under vector addition and scalar multiplication, is it not? That would mean it is a vector space, no?
 
vela said:
Not sure what that means.

That was a major mistake on my part of misinterpreting what he meant by "points into the first quadrant".
 
Retribution said:
It's closed under vector addition and scalar multiplication, is it not?

Is it?
Is [itex]\vec v_1 + a\vec v_2[/itex] still in that subset?
 

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