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Homework Help: Question about Linear dep/independence

  1. May 7, 2005 #1
    linear independece

    Hello. I have a problem here that i don't know how to start:

    Determine whether this set of vectors is Linearly dependent or independent.
    {u, v, w} where 4u-2v+3w = 0


    Any tips on how to begin proving whehter it's indep or dependent
     
    Last edited: May 7, 2005
  2. jcsd
  3. May 7, 2005 #2

    Hurkyl

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    Do you recall the definition of linear dependence?
     
  4. May 7, 2005 #3
    I know that a set is lin dependent if:
    -a set contains a zero vector
    -Let V1,V2,V3...Vr be vectors in Rn. If r > n, then set is dependent
    -one of the vectors is a lin combo of remaining vectors in a set
     
  5. May 7, 2005 #4

    Hurkyl

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    The actual definition, as I recall, is this:

    A set of vectors [itex]x_1, x_2, \ldots, x_n[/itex] is linearly dependent if and only if there exists scalars [itex]a_1, a_2, \ldots, a_n[/itex] (that are not all zero) such that [itex]a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0[/itex].

    Or, equivalently,

    A set of vectors [itex]x_1, x_2, \ldots, x_n[/itex] is lienarly independent if [itex]a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = 0[/itex] implies that each of the [itex]a_i[/itex] are zero.

    (Not only is it the definition I recall, but one of the more useful characterizations of linear dependence!)


    So the problem is fairly trivial. :smile: But it's still straightforward using the conditions you listed... for example, can you find a way to write u as a linear combination of v and w?
     
  6. May 7, 2005 #5
    Those are theorems derived from the definition... the definition will give you the answer directly.

    edit: too late :D
     
  7. May 7, 2005 #6
    a set of vectors [tex]{v_1, v_2,...v_k}[/tex] is linearly independent <==> [tex]a_1 v_1 + a_2 v_2 + ... + a_k v_k = 0[/tex] implies all the [tex]a_i = 0[/tex]
    that's the most basic definition of linear independence i learned. i don't really know what the problem is though. wouldn't the dimension of u, v, w matter?
     
  8. May 7, 2005 #7
    I figured it out. Thank so much
     
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