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

[twoflower]

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:

$$0 \in W$$

$$a \in W, b \in W \rightarrow a + b \in W$$

$$a \in \mathbb{K}, v \in W \rightarrow a.v \in W$$

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.

 

[Muzza]

It's correct.

 

[twoflower]

It's correct.

Just for the safety's sake - you mean my conclusion is correct or the statement is correct?

 

[Muzza]

Oh, didn't see that ambiguity. ;) I mean that your conclusion was correct.

 

[mathwonk]

also your argument is correct.