1. ## Vectors and linear independence. help please.

(a) If vectors u, v and w are linearly independent, will (u+v), (v+w) and (u+w) also be linearly independent? Justify your answer.

(b)If vectors u, v and w are linearly independent, will (u-v), (v-w) and (u-w) also be linearly independent? Justify your answer.

The answer in the back of the book is (a) yes. (b) no

Any help on why this is?

Cheers

2. Originally Posted by surfer101
(a) If vectors u, v and w are linearly independent, will (u+v), (v+w) and (u+w) also be linearly independent? Justify your answer.

(b)If vectors u, v and w are linearly independent, will (u-v), (v-w) and (u-w) also be linearly independent? Justify your answer.

The answer in the back of the book is (a) yes. (b) no

Any help on why this is?

Cheers
First recall what it means if vectors u, v and w are linearly independent. Now consider:

(a) $\displaystyle \alpha (u + v) + \beta (v + w) + \gamma (u + w) = (\alpha u + \beta v + \beta w) + (\gamma u + \alpha v + \gamma w)$ ....

(b) $\displaystyle \alpha (u - v) + \beta (v - w) + \gamma (u - w) = (\alpha u + \beta v + \beta w) - (\gamma u + \alpha v + \gamma w)$ ....

where $\displaystyle \alpha$, $\displaystyle \beta$ and $\displaystyle \gamma$ are scalars.