Can a Basis for a Subspace Always Include Specific Elements of a Larger Basis?

  • Context: Graduate 
  • Thread starter Thread starter ehsan_thr
  • Start date Start date
  • Tags Tags
    Basis Subspace
Click For Summary

Discussion Overview

The discussion revolves around the question of whether a basis for a subspace \( w \) of a vector space \( v \) can always include specific elements from a larger basis of \( v \). The context includes theoretical considerations about vector spaces, bases, and subspaces, as well as implications for linear independence and orthogonality.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that it is not always possible to find a basis for the subspace \( w \) that includes \( m \) elements from the basis of \( v \), citing examples such as the set of linear polynomials.
  • Others argue that while a basis for \( w \) may not include elements from the basis of \( v \), any basis for \( w \) can be extended to form a basis for \( v \).
  • A participant seeks to prove that for every proper subspace \( W \) of a finite-dimensional inner product space \( V \), there exists a non-zero vector \( x \) that is orthogonal to \( W \).
  • Another participant suggests starting with an orthogonal basis of \( W \) and using a vector from \( V \) that is not in \( W \) to find a vector orthogonal to \( W \).

Areas of Agreement / Disagreement

Participants generally disagree on whether a basis for the subspace \( w \) can include specific elements from the basis of \( v \). Multiple competing views remain regarding the relationship between bases of subspaces and their parent vector spaces.

Contextual Notes

Limitations include the dependence on specific examples and definitions of bases and subspaces, as well as unresolved mathematical steps related to orthogonality and linear combinations.

ehsan_thr
Messages
3
Reaction score
0
suppose that {ei} 1<=i<=n is a basis for v and w is a subspace of v of dimension m<n ... . so Can we always find a basis for w which includes m elements of {ei} 1<=i<=n ?

does always w include m elements of {ei} 1<=i<=n ?
 
Physics news on Phys.org
What does it mean for a list of vectors to be the basis for a vector space V ?
 
It means ,every vector of v is a linear combination of elements of the basis ... . ( The elements of v must be linear independent )
I have tried to prove my question , but i coudn't till now ... . my ultimate goal is to prove that for Every proper subspace W of a finite dimensional inner product space V there is a non zero vector x which is orothogonal to w ... .
 
ehsan_thr said:
suppose that {ei} 1<=i<=n is a basis for v and w is a subspace of v of dimension m<n ... . so Can we always find a basis for w which includes m elements of {ei} 1<=i<=n ?

does always w include m elements of {ei} 1<=i<=n ?

No. For example, let v be the set of linear polynomials and let {ei}= {x- 1, x+ 1}. Let w be the set of all multiples of x. Then w is a one dimensional subspace of v and any basis for w must be {ax} for some non-zero number a. None of those is in {ei}.

The other way is true: given any basis for subspace w we can extend it to a basis for v.
 
No. For example, let v be the set of linear polynomials and let {ei}= {x- 1, x+ 1}. Let w be the set of all multiples of x. Then w is a one dimensional subspace of v and any basis for w must be {ax} for some non-zero number a. None of those is in {ei}.

The other way is true: given any basis for subspace w we can extend it to a basis for v.
T 01:59 PM

Thank you , so much , i will find another solution for my problem ... .
 
Here's how I might approach the problem:

Start with an orthogonal basis of W, and pick any vector v in V that's not in W. Using your basis for W, can you use v to find a vector that's orthogonal to W?
 

Similar threads

Undergrad Vector subspace and basis vectors in the context of data science
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
4K
MHB Finding a Basis for a Linear Subspace Orthogonal to a Given Point P in R^3
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Non orthogonal basis and the lines of its coordinate grid
  • · Replies 9 ·
Replies
9
Views
4K
Graduate Confusion about the Moyal-Weyl twist
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Standard basis and other bases...
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Orthogonal basis to find projection onto a subspace
  • · Replies 5 ·
Replies
5
Views
5K
MHB Intersection of all subspace of V is the empty set
  • · Replies 8 ·
Replies
8
Views
2K
MHB Show that there are vectors to get a basis
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Basis of a Subspace of a Vector Space
  • · Replies 8 ·
Replies
8
Views
3K