Question

Is there a difference between a vector space and a subspace. I understand the concept that the subspace would be a box inside a bigger box (the vector space) but are their properties the same, or do they only share part of them ? I don't see why they would not share all of their properties, but my notes are vague and seems to indicate that they aren't.

Answer 1

a vector space have has ten axioms that must be fullfiled inorder for it to be a vector space

Answer 2

yep that I understand, I have the axioms right in front of my eyes. But later in my notes, my teacher starts to mention subspaces, give a couple of properties, which seems to be the same as the one from the vectors spaces, he dosen't specify if the subspace must fullfil the same properties/axioms in order to exist.

Answer 3

well, a vector space can be formed from subsets of other vector spaces, and those subsets are then called subspaces

Answer 4

about the couple of properties i'm reffering to, it's only like 4 properties

Answer 5

mainly because these (subspaces) fullfill three properties, closure under scalar multiplication, closure under addition, and zero vector is in the subspace

Answer 6

so if I get what you're saying, yes, the subspaces and vector spaces have the same properties ?

Answer 7

yes, because of this: if the subspace satisfies those three properties(mentioned above), then it itself is a vector space, which of course satifies all then axioms

Answer 8

ten*

Answer 9

awesome ! But in case the subspace does not satisfy the three properties, is it still considered a subspace at all ? I mean, if I have vector space V and I want to check if subspace V spans it, and I find out that it is not closed under vector addition or scalar multiplication or the zero vector is not part of it, could it still be a subspace of an other vector space ? And also, one thing I'm not sure, and I might be making you repeat the same things (i'm sorry if I am), if the subspace is closed under addition/vector mult. and contains the zero vector, then it is a vector space, can it still be subspace, or once it reaches this point, its a vector space of its own ?

Answer 10

*subspace H not subspace V

Answer 11

yes the vector can still be a subspace of a vector space, but we also consider it a vector space. If the subspace does not statisfy those three properites then it is not a subspace of V

Answer 12

ok but if its not a subspace of V can it be a subspace of an other vector space, let's say W, with different restrictions ?

Answer 13

I guess not... since we absolutely need the three restriction to agree with the 10 axioms

Answer 14

yeah sure, it could be a subspace of another vector space, but if we have a vector space (and we must remeber that a vector space has to satifiy the ten axioms) then a subspace of a particular vector space has to satisfy the three axioms mentioned

Answer 15

Ahhhhh !! It makes so much more sense now !! Thank you so much for your time, it's very appreciated ! :)