Assume that $v_1,v_2 \space and \space v_3$ are linearly independent.

I have a function: $c_1v_1 + c_2(v_1+v_3) + c_3(v_2-v_1) = 0$

How can I show that $c_1 = c_2 = c_3 = 0?$

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    $\begingroup$ You cannot, unless you know something about $v_i$'s. $\endgroup$ – Kavi Rama Murthy Sep 15 '20 at 10:02
  • $\begingroup$ I think I have to find some vectors v1,v2,v3 that can proof that. The question is how. $\endgroup$ – Sachihiro Sep 15 '20 at 10:04
  • $\begingroup$ Any set of three linearly independent vectors will do. $\endgroup$ – Kavi Rama Murthy Sep 15 '20 at 10:07
  • $\begingroup$ It seems you are assuming that $v_1, v_2, v_3$ are linearly independent and want to show that also $v_1$, $v_1+v_3$ and $v_2-v_1$. You should clarify that in your question. $\endgroup$ – user Sep 15 '20 at 10:10
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    $\begingroup$ @user yea that is correct. I apologize, i am new to linear algebra. $\endgroup$ – Sachihiro Sep 15 '20 at 10:14

By definition, assuming that $v_1, v_2, v_3$ are linearly independent vectors, we have that

$$c_1v_1 + c_2(v_1+v_3) + c_3(v_2-v_1) = 0 \iff (c_1+c_2-c_3)v_1+c_3v_2+c_2v_3=0$$

and we obtain

  • $c_1+c_2-c_3=0$
  • $c_3=0$
  • $c_2=0$

that is $c_1=c_2=c_3=0$ therefore also $w_1=v_1$, $w_2=v_1+v_3$ and $w_3=v_2-v_1$ are linearly independent vectors.

  • $\begingroup$ A naive question, do I have to provide vectors v1,v2,v3 to prove? $\endgroup$ – Sachihiro Sep 15 '20 at 10:46
  • $\begingroup$ @Sachihiro The proof holds for any set of three given independent vectors. $\endgroup$ – user Sep 15 '20 at 11:08

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