Difference between a vector subspace and subset?

Click For Summary
A vector subspace is a vector space contained within another vector space, characterized by being closed under vector addition and scalar multiplication. In contrast, a vector subset may not satisfy these properties. The discussion centers on whether the subset of vectors pointing into the first quadrant qualifies as a vector subspace. It is concluded that this subset does not form a vector subspace because it does not include all necessary components, particularly those not confined to the first quadrant. The confusion arises from misinterpreting the conditions required for a subset to be classified as a vector subspace.
-Dragoon-
Messages
308
Reaction score
7
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.
 
Physics news on Phys.org
Consider the subset of all vectors that point into the first quadrant.
Is that a vector subspace?
 
robphy said:
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?
 
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?
 
Retribution said:
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.
 
robphy said:
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?
 
vela said:
Not sure what that means.

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

Similar threads

Vector Subspaces: Determining U as a Subspace of M4x4 Matrices
  • · Replies 8 ·
Replies
8
Views
2K
Help with linear algebra: vectorspace and subspace
  • · Replies 15 ·
Replies
15
Views
3K
Determining if a subset W is a subspace of vector space V
  • · Replies 8 ·
Replies
8
Views
2K
Proving that a subset is a subspace
  • · Replies 4 ·
Replies
4
Views
2K
Understanding Subspaces: Criteria and Examples (R3, Integers, R2)
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Vector subspace and basis vectors in the context of data science
  • · Replies 6 ·
Replies
6
Views
3K
[Linear Algebra] Another question on subspaces
  • · Replies 15 ·
Replies
15
Views
2K
Determine whether the subset is a vector subspace
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Smallest subspace if a plane and a line are passing through the origin
  • · Replies 2 ·
Replies
2
Views
2K
[Linear Alg] Determining which sets are subspaces of R[x]
  • · Replies 7 ·
Replies
7
Views
2K