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

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In summary, the conversation discusses whether the statement "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" is correct or not. The participants agree that V \ U cannot be a subspace of V in general, as it may not fulfill the necessary conditions for a subspace. They also confirm that the conclusion and argument presented are correct.
  • #1
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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  • #2
It's correct.
 
  • #3
Muzza said:
It's correct.
Just for the safety's sake - you mean my conclusion is correct or the statement is correct? :smile:
 
  • #4
Oh, didn't see that ambiguity. ;) I mean that your conclusion was correct.
 
  • #5
also your argument is correct.
 

1. What is a subspace?

A subspace is a subset of a vector space that also satisfies the properties of a vector space. This means that it must contain the zero vector, be closed under vector addition and scalar multiplication, and must follow the axioms of vector addition and scalar multiplication.

2. How do you determine if V \ U is a subspace of V?

To determine if V \ U is a subspace of V, we need to check if it satisfies the properties of a vector space. This means checking if the zero vector is in V \ U, if it is closed under vector addition and scalar multiplication, and if it follows the axioms of vector addition and scalar multiplication.

3. What is the difference between V \ U and V?

V \ U is the set of all elements in V that are not in U. This means that V \ U is a subset of V, and it may have different properties than V. V, on the other hand, is the original vector space and contains all of its elements without any exclusions.

4. Can V \ U be a subspace of V if U is also a subspace of V?

Yes, it is possible for V \ U to be a subspace of V even if U is also a subspace of V. This is because V \ U is a subset of V, and it may have different properties than U. As long as V \ U satisfies the properties of a vector space, it can be considered a subspace of V.

5. How can you prove that V \ U is a subspace of V?

To prove that V \ U is a subspace of V, we need to show that it satisfies the properties of a vector space. This can be done by checking if the zero vector is in V \ U, if it is closed under vector addition and scalar multiplication, and if it follows the axioms of vector addition and scalar multiplication. If all of these conditions are met, then we can conclude that V \ U is a subspace of V.

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