The discussion revolves around proving that a vector space cannot be the union of two proper subspaces. Participants are exploring the definitions and implications of proper subspaces within the context of vector spaces.
There is an active exploration of definitions and potential misunderstandings regarding the properties of proper subspaces. Some participants have offered insights into the relationship between unions and spans, suggesting that the union of two subspaces may not maintain the properties of a subspace unless one is contained within the other.
Participants note that the original poster did not include the definition of a proper subspace, leading to confusion about the assumptions necessary for the proof. There are indications that the problem may require clarification on the nature of the subspaces involved.
Mark44 said:What is the definition of "proper subspace"? You should have included that definition when you posted the problem.
Mark44 said:But how is this term defined? What you gave is not the definition.
Coto said:There seems to be something missing in this problem. For example, take the case where U,W are proper subspaces of V, and U \subset W, then U \cap W \equiv W is also a proper subspace of V.
Are you sure there's not some other restrictions regarding the subspaces?
That was a typo. Coto meant U \cup W = W.Dustinsfl said:Would it matter that it is the union and you have written the intersection?
jambaugh said:I think the issue is that the union is not the same as the span. The union of two subspaces will not be a subspace unless one of the subspaces is contained within the other (is a subspace of the subspace) in which case the union is the larger subspace.
So apply that to the question at hand...