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Homework Help: Linearly independent

  1. Feb 28, 2010 #1
    Let x1, x2,...,xk be linear independent vectors in a vector space V.

    If we delete a vector, say xk, from the collection, will we still have a linearly independent collection of vectors? Explain.

    By deleting a vector from linearly independent span, the other vectors, I believe, will remain independent; however, I don't know how to prove it.
     
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  3. Feb 28, 2010 #2

    Dick

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    Why don't you start with the definition of linearly independent? That's usually a good strategy.
     
  4. Feb 28, 2010 #3
    Well vectors are lin. ind. if the the det doesn't 0 and if all coefficients are 0. Since I know they are already ind., deleting one shouldn't change the coefficients but I don't know how to set it up in a proof still.
     
  5. Feb 28, 2010 #4

    Dick

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    det of what? I don't think that has much to do with the definition of linearly independent. Does it? I suggest you look it up. State it clearly.
     
  6. Feb 28, 2010 #5
    Determinant of the vectors in the span doesn't equal 0 then they are linearly ind.
     
  7. Feb 28, 2010 #6

    Dick

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    Baloney. Determinant is only defined for a square matrix. That's a special case. There's a much more general definition of linear independence.
     
  8. Feb 28, 2010 #7
    What does it mean if x1, x2, x3 are linearly independent? It means that the solution to a1x1 + a2x2 + a3x3 = 0 is ai = 0 for all i=1,2,3. Apply this definition to k vectors.

    Now, does this still hold if you take out some vector in {x1,...., xk}? Remove some xi from the set and construct the equation I did above. Does it follow that all the ai's are 0?
     
  9. Feb 28, 2010 #8
    If you remove a vector, the other coefficients should still remain the same = 0.
     
  10. Mar 1, 2010 #9

    HallsofIvy

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    Yes, and therefore what?

    VeeEight is suggesting a proof by contradiction.
     
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