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Basis of subspace

  1. Feb 8, 2009 #1
    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 ????
     
  2. jcsd
  3. Feb 8, 2009 #2
    What does it mean for a list of vectors to be the basis for a vector space V ?
     
  4. Feb 8, 2009 #3
    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 ... .
     
  5. Feb 8, 2009 #4

    HallsofIvy

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    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.
     
  6. Feb 8, 2009 #5
    Thank you , so much , i will find another solution for my problem ... .
     
  7. Feb 8, 2009 #6
    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?
     
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