Union of subspaces of a linear space

  • Thread starter de_brook
  • Start date
  • #1
74
0

Main Question or Discussion Point

Is there a linear space V in which the union of any subspaces of V is a subspace except the trivial subspaces V and {0}? pls help
 

Answers and Replies

  • #2
316
0
A vector space V can only have non-trivial subspaces if [tex]\dim V\ge 2[/tex].
This means you can choose two linearly independent vectors u, w, which generate 1-dimensional subspaces U, W respectively. Can [tex]U\cup W[/tex] be a subspace? Hint: try to find a linear combination of u,w that is not in [tex]U\cup W[/tex].
 
  • #3
74
0
A vector space V can only have non-trivial subspaces if [tex]\dim V\ge 2[/tex].
This means you can choose two linearly independent vectors u, w, which generate 1-dimensional subspaces U, W respectively. Can [tex]U\cup W[/tex] be a subspace? Hint: try to find a linear combination of u,w that is not in [tex]U\cup W[/tex].
I have tried searching for such spaces but i could only find for spaces whose dimension is less than 2.
 
  • #4
316
0
I have tried searching for such spaces but i could only find for spaces whose dimension is less than 2.
[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
 
  • #5
74
0
[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
yyat; Do you mean i can obtain a linear subspace V of \mathbb{R}^n such that the union of any subspaces of V is a subspace of V?
 
  • #6
74
0
[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
yyat; Do you mean i can obtain a linear subspace V of [tex]\mathbb{R}^n such that the union of any subspaces of V is a subspace of V?
 
  • #7
74
0
[tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
yyat; Do you mean i can obtain a linear subspace V of [tex]\mathbb{R}^n[/tex], such that the union of any subspaces of V is a subspace of V?
 
  • #8
34
0
I think the short answer is No.
 
  • #9
74
0
what about if you are not working n dimensional space, can you still find such a space
 
  • #10
74
0
I think the short answer is No.
what about if you are not working with n-dimensional space, can you still find such a space?
 
  • #11
34
0
If the dimension of the space is less than two then the only subspace are V and {0} as yyat pointed out. Hence your question is answered in this case.

If the dimension of the space is greater or equal to two then consider spaces X and Y generated by linearly independent vectors x and y. x+y does not belong to [tex] X \Cup Y [/tex]. Implying you can't pick any subspaces and the union will be a subspace.
 
  • #12
74
0
If the dimension of the space is less than two then the only subspace are V and {0} as yyat pointed out. Hence your question is answered in this case.

If the dimension of the space is greater or equal to two then consider spaces X and Y generated by linearly independent vectors x and y. x+y does not belong to [tex] X \Cup Y [/tex]. Implying you can't pick any subspaces and the union will be a subspace.
thanks so much.
 

Related Threads on Union of subspaces of a linear space

  • Last Post
Replies
5
Views
3K
Replies
1
Views
2K
Replies
4
Views
2K
Replies
8
Views
1K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
4
Views
19K
  • Last Post
Replies
22
Views
773
  • Last Post
2
Replies
28
Views
4K
Replies
8
Views
3K
Top