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Linear dependency

Yankel

Active member
Jan 27, 2012
398
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
Jan 26, 2012
995
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.