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

  • Thread starter ashina14
  • Start date
  • Tags
    Subspace
In summary, a user named hedipaldi has been warned multiple times for posting complete solutions on Physics Forums, and has continued to do so despite these warnings. Please refrain from posting complete solutions in the future.
  • #1
ashina14
34
0

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...
 
Physics news on Phys.org
  • #2
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
How do I show that the span of B is appropriate for T?
 
  • #4
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
i'm not sure if i need to quote the theorem
 
  • #6
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
How do I prove that? I don't see an obvious connection here
 
  • #8
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
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
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
Thanks for the help anyway :)
 
  • #12
Mark44 said:
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.

ashina14 said:
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
hedipaldi said:
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!
 

1. How do you prove the existence of a subspace T within a given subspace S?

In order to prove the existence of a subspace T within a given subspace S, we must show that every vector in S can be written as a sum of two vectors, one from S and one from T, and that the intersection of S and T is only the zero vector.

2. What is the significance of the direct sum symbol in the statement "V=S⊕T"?

The direct sum symbol, ⊕, signifies that the two subspaces S and T are mutually orthogonal, meaning that the only vector they have in common is the zero vector. It also shows that together, S and T span the entire vector space V.

3. Can you provide an example of a subspace S and its corresponding subspace T for which V=S⊕T holds true?

Yes, for example, let V be the set of all 2x2 matrices, S be the set of all diagonal matrices, and T be the set of all upper triangular matrices. Then, V=S⊕T because every matrix in V can be written as the sum of a diagonal matrix from S and an upper triangular matrix from T.

4. How does the existence of a subspace T within S affect the dimension of S and T?

The dimension of S and T are not affected by the existence of T within S. The dimension of S remains the same, while the dimension of T is determined by the number of vectors needed to span the orthogonal complement of S within V.

5. Is the existence of a subspace T within S unique or can there be multiple subspaces T that satisfy V=S⊕T?

The existence of a subspace T within S is not unique. In fact, there can be infinitely many subspaces T that satisfy V=S⊕T. This is because there can be infinitely many combinations of vectors from S and T that can span V and satisfy the conditions of being mutually orthogonal and only intersecting at the zero vector.

Similar threads

  • Calculus and Beyond Homework Help
Replies
0
Views
418
  • Calculus and Beyond Homework Help
Replies
1
Views
421
  • Calculus and Beyond Homework Help
Replies
2
Views
266
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
24
Views
672
  • Calculus and Beyond Homework Help
Replies
14
Views
531
  • Calculus and Beyond Homework Help
Replies
4
Views
953
  • Calculus and Beyond Homework Help
Replies
5
Views
5K
  • Calculus and Beyond Homework Help
Replies
8
Views
558
Back
Top