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Union of subspaces of a linear space

  1. Mar 7, 2009 #1
    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
     
  2. jcsd
  3. Mar 7, 2009 #2
    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].
     
  4. Mar 7, 2009 #3
    I have tried searching for such spaces but i could only find for spaces whose dimension is less than 2.
     
  5. Mar 7, 2009 #4
    [tex]\dim\mathbb{R}^n=n[/tex], surely you knew that?
     
  6. Mar 7, 2009 #5
    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?
     
  7. Mar 7, 2009 #6
    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?
     
  8. Mar 7, 2009 #7
    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?
     
  9. Mar 7, 2009 #8
    I think the short answer is No.
     
  10. Mar 7, 2009 #9
    what about if you are not working n dimensional space, can you still find such a space
     
  11. Mar 7, 2009 #10
    what about if you are not working with n-dimensional space, can you still find such a space?
     
  12. Mar 7, 2009 #11
    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.
     
  13. Mar 7, 2009 #12
    thanks so much.
     
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