Question about Linear dep/independence

  • Thread starter EvaBugs
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  • #1
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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
 
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Answers and Replies

  • #2
Hurkyl
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Do you recall the definition of linear dependence?
 
  • #3
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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
 
  • #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?
 
  • #5
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Those are theorems derived from the definition... the definition will give you the answer directly.

edit: too late :D
 
  • #6
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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?
 
  • #7
19
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I figured it out. Thank so much
 

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