suppose that [tex]\{ v_1,v_2,v_3\}[/tex] is a set of dependent vectors. Prove that [tex] \{v_1,(v_1+v_2),v_3\}[/tex] is also a set of dependent vectors?(adsbygoogle = window.adsbygoogle || []).push({});

Ok, I have two ways of going about it. My concern is about the first one, it looks like it is not correct the way i did it, i think so, cuz i managed to find some kind of counter example, but i cannot see why it doesn't hold,because it looks logical to me.

Method 1.

Since [tex]\{ v_1,v_2,v_3\}[/tex] lin. dependent set, we know that the vector equation:

[tex]av_1+bv_2+cv_3=0[/tex] has a non trivial solution, that is at least one of a,b,c is a non-zero scalar. Now, let's take a linear combination of the vectors [tex] \{v_1,(v_1+v_2),v_3\}[/tex]

[tex] av_1+b(v_1+v_2)+cv_3=0[/tex] my goal is to show that this eq. has a nontrivial sol. that is that one of, a,b,c must be nonzero. So:

[tex] av_1+b(v_1+v_2)+cv_3=0=>av_1+bv_1+bv_2+cv_3=0=>(av_1+bv_2+cv_3)+bv_1=0=>0+bv_1=0[/tex] SO, i concluded then that [tex]v_1=0, or, b=0[/tex] Then if [tex] v_1=0[/tex] we know that every set that contains the zero vector is dependent, so i concluded that the set [tex] \{v_1,(v_1+v_2),v_3\}[/tex] is also dependent. On the other hand, if b=0, then either a, or c must be different from zero, so again this set is lin. dependent.

I am not convinced that this works because of this:

Let[tex]v_1=[1,2,3]^T,v_2=[2,-1,4]^T,v_3=[0,5,2]^T[/tex] then none of these vectors is zero, and they are also lin. dependent. so if i try to follow the same logic as above, in here to show that the set [tex] \{v_1,(v_1+v_2),v_3\}[/tex] is lin. dependent i end up with a contradiction:

[tex] bv_2=0[/tex] but, b is not necessarly zero. SO, where have i gone wrong? I mean why the above doesn't work?

Here is the second method that i used to prove that, this is way shorter, I again took a lin. comb on the vectors in the set:[tex] \{v_1,(v_1+v_2),v_3\}[/tex]

THat is:[tex] av_1+b(v_1+v_2)+cv_3=0[/tex] then after a little manipulation we have:

[tex] (a+b)v_1+bv_2+cv_3=0[/tex] now since [tex]v_1,v_2,v_3[/tex] are lin.dependent then one of a+b, b, c must be different from zero. SO, we can automatically conclude that

:[tex] av_1+b(v_1+v_2)+cv_3=0[/tex] has nontrivial sol, hence it it dependent set.

Please enlighten me, i am kind of confused?

Thnx in advance!

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# Linearly dependent set of vectors?

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