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Linear independence proof

  1. Jan 22, 2006 #1
    Hi I just need some help on understanding some general notation in this quesiton:

    Prove if {x_1,x_2,..,x_m} is linearly independent then so is {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m}.

    I don't really understand what the difference between {x_1,x_2,...,x_i-1, x_i+1,...,x_m} for every i in {1,2,...,m} and {x_1,x_2,..,x_m} is.

    Any help clarifying this would be great, and any hints for the question would be must appreciated, thanks.
  2. jcsd
  3. Jan 23, 2006 #2

    matt grime

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    The second set omits the i'th vector.

    Eg the large set is {a,b,c,d} and there are 4 other sets: {b,c,d}, {a,c,d}, {a,b,d}, {a,b,c}
  4. Jan 23, 2006 #3


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    If you understand that, you can easily prove this by contradiction.
    Suppose one of the smaller sets is linearly dependent, then one of its elements is a lineair combination of the others. What does that tell you about the larger set then?
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