Difference between a vector subspace and subset?

[-Dragoon-]

This is quite confusing to me. I know a vector subspace is a vector space within another vector space and is closed under the operations of the vector space it lies in, but how exactly does it differ from vector subsets? Anyone care to explain or clarify this? My textbook is completely terrible at explaining this. Thanks in advance.

 

[robphy]

Consider the subset of all vectors that point into the first quadrant.
Is that a vector subspace?

 

[-Dragoon-]

Consider the subset of all vectors that point into the first quadrant.
Is that a vector subspace?

So, in this case, the vector space V would be the entire first quadrant. Since the vectors that point into the first quadrant would imply they have components not in the first quadrant, that would mean that it is not a vector subspace, correct?

 

[robphy]

Take a 2D vector space (a plane).
The subset of all vectors [with base point at the origin] that point into the first quadrant have positive x- and y-components. Do those vectors satisfy the properties of a vector space?

 

[vela]

So, in this case, the vector [STRIKE]space[/STRIKE] subset V would be the entire first quadrant. Since the vectors that point into the first quadrant would imply they have components not in the first quadrant, that would mean that it is not a vector subspace, correct?

Not sure what that means.

 

[-Dragoon-]

Take a 2D vector space (a plane).
The subset of all vectors [with base point at the origin] that point into the first quadrant have positive x- and y-components. Do those vectors satisfy the properties of a vector space?

It's closed under vector addition and scalar multiplication, is it not? That would mean it is a vector space, no?

 

[-Dragoon-]

Not sure what that means.

That was a major mistake on my part of misinterpreting what he meant by "points into the first quadrant".

 

[robphy]

It's closed under vector addition and scalar multiplication, is it not?

Is it?
Is $\vec v_1 + a\vec v_2$ still in that subset?