The discussion revolves around proving whether a field F has only two subspaces: the zero subspace {0} and the field itself F. Participants explore the properties of subspaces in the context of vector spaces and question the definitions involved.
The discussion is active, with various lines of reasoning being explored. Some participants are providing insights into the properties of subspaces, while others are seeking clarification on definitions and dimensions. There is no explicit consensus yet, but the conversation is progressing towards understanding the relationship between subspaces and fields.
Participants note that they have not yet covered dimensions in their coursework, which may affect their ability to engage fully with the problem. The definitions of subspaces and their properties are under scrutiny, particularly in relation to fields viewed as vector spaces over themselves.
Well, yes you can, but that would just prove that the empty set and the field itself are subspaces. It wouldn't prove that those are the only two subspaces of F.Samuelb88 said:Not sure on this one, but... Can you show that the zero subspace, and the field itself has: (a) the "zero object" of F exists in both subspaces. (b) Each subspace is closed under "object" addition. (c) Each subspace is closed under scalar multiplication. ? Just a thought.
We haven't gotten over dimensions in my class so I don't know that to be honest. Do you know a link that can direct me to know more about this, or do you have the time to explain it? It would seem pretty easy once I know this piece of information.Office_Shredder said:If F is a vector space over itself, what dimension is it?
Then remember any proper subspace has to have a smaller dimension
Petek said:Suppose that U is a non-zero subspace of F. Then U is closed under scalar multiplication. Let x be any non-zero element of U and let y be any element of F. Can you find a scalar multiple of x that equals y? What does that tell you?
Petek
Yes.Anonymous217 said:Is it supposed to show that any non-zero subspace of F must be F itself?
I can understand the rest of the proof from then on, but how do you show that U must be = F? What scalar multiple of x equals y?
Anonymous217 said:Is it supposed to show that any non-zero subspace of F must be F itself? I can understand the rest of the proof from then on, but how do you show that U must be = F? What scalar multiple of x equals y?
HallsofIvy said:Do you mean subfield rather than subspace? I don't know what it means to say that a field has a subspace.