Why Does a Subset of a Vector Space Need the Zero Vector to Be a Subspace?

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Discussion Overview

The discussion revolves around the requirement for a subset of a vector space to include the zero vector of the original vector space in order to qualify as a subspace. Participants explore the intuition behind this requirement and the implications of having different zero vectors in subspaces.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about why a subspace must include the zero vector of the original vector space, suggesting it is not intuitive.
  • Another participant argues that a subspace must be closed under addition and scalar multiplication, leading to the conclusion that the zero vector must be included in the subspace.
  • A different viewpoint suggests that if subspaces had their own zero vectors, combining them could lead to inconsistencies, such as disappearing elements and discontinuities.
  • One participant reiterates the initial question about the intuition behind the requirement for the zero vector and asks for alternatives.
  • Another participant questions the idea of a subspace that is not a vector space, asserting that a subspace must possess vector space properties.
  • A final contribution emphasizes that the term "subspace" inherently implies a vector space, distinguishing it from a mere subset.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the intuitiveness of the requirement for the zero vector. There are competing views regarding the implications of having different zero vectors in subspaces and the definition of a subspace itself.

Contextual Notes

Some participants highlight the need for clarity in definitions and the properties that subspaces must satisfy, indicating that the discussion may depend on specific interpretations of terms like "subspace" and "subset."

torquerotates
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I am curious as to why a subset of a vector space V must have the vector space V's zero vector be the subsets' zero vector in order to be a subspace. Its just not intuitive.
 
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Why would it not be intuitive? A subspace, U, of vector space V must be closed under addition and scalar multiplication. If v is in subspace U, the (-1)v = -v is also. Then v+ (-v)= 0 is in U. That is, the zero vector of V, 0, is in U. But it is easy to show that the zero vector of a unique. Since 0 is in U and, of course, v+ 0= v for all v in U, there cannot be another zero vector in U.
 
If each subspace has its own zero vector, then combine these subspaces in order to get a bigger subspace or even the whole space. We will get bunch of different zeros and the whole space will very entertaining, suddenly disappearing elements and discontinuities...

Also note that these are the rules of the game that are required to have, rather than anticipating their existence based on intuition.
 
torquerotates said:
I am curious as to why a subset of a vector space V must have the vector space V's zero vector be the subsets' zero vector in order to be a subspace. Its just not intuitive.

What would you suggest as an alternative?
 
ejungkurth said:
What would you suggest as an alternative?

A subspace which isn't a vector space?
 
by definition a subspace have to be a vector space, and then all the other peoples arguments holds. What you are sugesting is just a simple subset, but that is not so interresting i linear algebra because it don't have the vector space properties.

by the way: The space in subspace means vectorspace, so it should really say subvectorspace. But a subset isn't a space so that's why there is no ambiguity.
 

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