Is V \ U a Subspace of V? Examining the Conditions for Subspace Inclusion

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Discussion Overview

The discussion revolves around the conditions under which the set difference V \ U, where V is a vector space and U is a subspace, can be considered a subspace of V. Participants are examining the validity of a statement regarding this relationship, referencing definitions and properties of subspaces.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant argues that V \ U cannot be a subspace because it does not contain the zero vector, which is a requirement for any subspace.
  • Another participant confirms that the initial conclusion regarding V \ U not being a subspace is correct.
  • Further clarification is sought regarding whether the correctness pertains to the conclusion or the original statement.
  • Subsequent responses affirm the correctness of the conclusion drawn by the first participant.

Areas of Agreement / Disagreement

Participants generally agree that the conclusion drawn about V \ U not being a subspace is correct, though there is some ambiguity regarding the interpretation of the original statement.

Contextual Notes

The discussion does not address potential exceptions or specific conditions under which V \ U might be a subspace, leaving those aspects unresolved.

twoflower
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Hi all,

this question was in a test the previous year:

Decide, whether this statement is right or not (in accord with the content of the lecture). Justify your decision:

Let V be a vector space and U its subspace. Then, in some cases V \ U could be the subspace of V, but generally it doesn't have to be a subspace of V

I think that V \ U can't be a subspace, because each subspace must fit this conditions:

[tex] 0 \in W[/tex]

[tex] a \in W, b \in W \rightarrow a + b \in W[/tex]

[tex] a \in \mathbb{K}, v \in W \rightarrow a.v \in W[/tex]

So, if U is subspace, it contains 0. So, V \ U doesn't contain 0 => it isn't a subspace.

Is this a right conclusion?

Thank you.
 
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It's correct.
 
Muzza said:
It's correct.
Just for the safety's sake - you mean my conclusion is correct or the statement is correct? :smile:
 
Oh, didn't see that ambiguity. ;) I mean that your conclusion was correct.
 
also your argument is correct.
 

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