Given a subspace S<=V, prove that there exists T<=V such that V=S⊕T.

  • Thread starter ashina14
  • Start date
  • #1
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Homework Statement



V is a vector space


The Attempt at a Solution



If S is smaller than V then there exists a T such that S + T = V. OTHERWISE S = V. I'm not sure what assumptions am I making which I could break down to prove...
 

Answers and Replies

  • #2
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choose a basis A for T,complete it to a basis for V by adding a set B of vectors.Now show that the span of B is appropriate for T.
 
  • #3
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How do I show that the span of B is appropriate for T?
 
  • #4
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I've come up with this. Does this seem right?

if S is the empty space, the solution is obvious
if dim S >= 1 there is a base of vectors (ui) of S.
And there is a theorem who says that
the family of vectors (ui) can be completed
with a family of vectors (vj) so that
the union (ui) with (vj) is a basis of V
finally the subspace T generated by (vj) = (v1, v2, ...)
in the complementary space of S so that
V=S⊕T
 
  • #5
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i'm not sure if i need to quote the theorem
 
  • #6
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Assume that v is a non zero vector in the intersection of S and T and prove that this contradicts the linear independence of the vectors in the union of A and B.
 
  • #7
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How do I prove that? I don't see an obvious connection here
 
  • #8
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hedipaldi,
It is against Physics Forums rules to post complete solutions. You have received numerous warnings about this, and each comes with a private message to you. Apparently you aren't reading your PMs so I am posting something to you in this thread.
 
  • #9
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I didn't ask for a complete solution, I'm genuinely stuck. I'm not that acquainted with the unusual rules here as I don't come here often. Each of your warning is about a separate issue and I don't repeat the same mistake again. I would appreciate if you understand my position.
 
  • #10
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I know this rule and indeed i answered by hints.however it was not understood so i tried to help more.I understand that this is unwanted and i will obey the rules of the forum.
sorry'
Hedi
 
  • #11
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Thanks for the help anyway :)
 
  • #12
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6,734
hedipaldi,
It is against Physics Forums rules to post complete solutions. You have received numerous warnings about this, and each comes with a private message to you. Apparently you aren't reading your PMs so I am posting something to you in this thread.

I didn't ask for a complete solution, I'm genuinely stuck. I'm not that acquainted with the unusual rules here as I don't come here often. Each of your warning is about a separate issue and I don't repeat the same mistake again. I would appreciate if you understand my position.
My post was addressed to hedipaldi, not you, ashina14. See above.

The rules are here: https://www.physicsforums.com/showthread.php?t=414380.
 
  • #13
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6,734
I know this rule and indeed i answered by hints.however it was not understood so i tried to help more.I understand that this is unwanted and i will obey the rules of the forum.
sorry'
Hedi
I'm glad to hear that!
 

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