Independent Subspace: Proving (or Disproving) Linear Independence

Click For Summary

Homework Help Overview

The discussion revolves around the concept of linear independence in vector spaces, specifically examining the relationships between different basis sets for those spaces. The original poster poses a question about the independence of two basis sets, B and C, in relation to another basis set D.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of linear independence and question the implications of having multiple basis sets for the same vector space. There is a discussion on whether the independence of sets B and D implies the independence of set C from D.

Discussion Status

The discussion is ongoing, with participants attempting to clarify definitions and relationships between the sets. Some have provided reasoning and examples to support their points, while others are questioning the assumptions made about the relationships between the basis sets.

Contextual Notes

There appears to be some ambiguity regarding the definitions of linear independence and the specific relationships between the basis sets, which may affect the clarity of the discussion.

hayu601
Messages
7
Reaction score
0
Suppose B = {b1,...,bn} and C={c1,...,cn} both are basis set for space V.
D = {d1,...,dn} is basis for space T.

If B and D is linearly independent, is C and D always independent too? How can we prove (disprove) it?
 
Physics news on Phys.org
I don't know what you mean by two sets of vectors being "independent". By saying that "B and D is linearly independent" do you mean that the set B\times D is a set of independent vectors in V\times T?
 
It means that every bi element B is not linear combination of vectors in D
 
If B and C are both separate basis for V, then C = aB.

And if B and D are linearly independent, Bb != D and thus, Bab = Cb != Da

So Cd != D for some scalar d=b/a

That's basically what you have to prove in a more elegant form.
 

Similar threads

  • · Replies 15 ·
Replies
15
Views
2K
  • · Replies 4 ·
Replies
4
Views
3K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 1 ·
Replies
1
Views
3K
  • · Replies 15 ·
Replies
15
Views
3K
  • · Replies 10 ·
Replies
10
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
Replies
1
Views
2K
  • · Replies 1 ·
Replies
1
Views
1K
Replies
5
Views
2K