# Linear dependency

#### Yankel

##### Active member
Hello,

I need some help with this one...any guidance will be appreciated.

In a vector space V there are 3 linearly independent vectors e1,e2,e3. Prove that the vectors:

e1+e2 , e2-e3 , e3+2e1

are also linearly independent.

Thanks...

#### Chris L T521

##### Well-known member
Staff member
Hello,

I need some help with this one...any guidance will be appreciated.

In a vector space V there are 3 linearly independent vectors e1,e2,e3. Prove that the vectors:

e1+e2 , e2-e3 , e3+2e1

are also linearly independent.

Thanks...
We want to see when the combination
$c_1(e_1+e_2)+c_2(e_2-e_3)+c_3(e_3+2e_1)=0$
Rearranging the terms we get
$(c_1+2c_3)e_1+(c_1+c_2)e_2+(c_3-c_2)e_3=0$
Since $e_1,e_2,e_3$ are independent, what does that say about the value of the coefficients $c_1+2c_3, c_1+c_2, c_3-c_2$?

You should have enough information now to finish off this problem.