# Thread: Linearly Independents Sums and Differences

1. ## Linearly Independents Sums and Differences

I am having trouble understanding the answers to the following two questions:

1. If vector u, v, and w are linearly independent, will u+v, v+w, and u+w also be linearly independent?
2. If vector u, v, and w are linearly independent, will u-v, v-w, and u-w also be linearly independent?

I know the answer to the first question is yes, and I know the answer to the second question is no, but I don't know why. Can someone provide me with an explanation. I can't come up with any justifications.

2. Originally Posted by 323k13l
I am having trouble understanding the answers to the following two questions:

1. If vector u, v, and w are linearly independent, will u+v, v+w, and u+w also be linearly independent?
2. If vector u, v, and w are linearly independent, will u-v, v-w, and u-w also be linearly independent?

I know the answer to the first question is yes, and I know the answer to the second question is no, but I don't know why. Can someone provide me with an explanation. I can't come up with any justifications.

3. Hello,

The definition of linearly independent vectors u, v & w is :

If there are a,b,c three scalars such as au+bv+cw=0, u, v and w are l-i if and only if a=b=c=0

What if a'(u+v)+b'(v+w)+c'(u+w)=0 ? Put the coefficients of u, v and w together. Are a', b' and c' necessarily null ?

The same for a'(u-v)+b'(v-w)+c'(u-w)=0, is there a case when a', b' and c' are not null, but where the sum is null ?

4. Thank you to both of you. It's obvious now, and it makes sense.