How to prove the linear independence of vectors in $\mathbb{R}^{3}.$

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 $$c_1 = c_2 = c_3 = 0?$$

• You cannot, unless you know something about $v_i$'s. – Kavi Rama Murthy Sep 15 '20 at 10:02
• I think I have to find some vectors v1,v2,v3 that can proof that. The question is how. – Sachihiro Sep 15 '20 at 10:04
• Any set of three linearly independent vectors will do. – Kavi Rama Murthy Sep 15 '20 at 10:07
• 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. – user Sep 15 '20 at 10:10
• @user yea that is correct. I apologize, i am new to linear algebra. – 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.

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