Subspace vs Subset: Understanding the Relationship

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Discussion Overview

The discussion revolves around the relationship between subspaces and subsets within the context of vector spaces. Participants explore whether a set must be a subset to qualify as a subspace, examining definitions and properties related to vector spaces and their subspaces.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • One participant questions if a set needs to be a subset to be a subspace of a vector space.
  • Another participant asserts that a subspace must indeed be a subset of the vector space.
  • Several participants clarify that a vector space consists of objects with defined operations of addition and scalar multiplication, and that a subspace inherits these operations as a subset.

Areas of Agreement / Disagreement

There appears to be agreement that a subspace must be a subset of a vector space, but the initial question indicates some uncertainty about this relationship.

Contextual Notes

The discussion does not address potential exceptions or specific definitions that may vary in different contexts.

Max.Planck
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Hi,

A quick question:
Does a set need to be a subset to be a subspace of some vector space?
 
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Max.Planck said:
Hi,

A quick question:
Does a set need to be a subset to be a subspace of some vector space?

Yes, a subspace of a vector space V necessarily needs to be a subset.
 
A vector space is a set of objects with "addition" and "scalar multiplication" defined. A "subspace" is a subset with the "inherited" addition and scalar multiplication.
 
HallsofIvy said:
A vector space is a set of objects with "addition" and "scalar multiplication" defined. A "subspace" is a subset with the "inherited" addition and scalar multiplication.

Thank you!
 

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