# Vectors and linear independence. help please.

• Mar 25th 2008, 03:04 AM
surfer101
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
• Mar 25th 2008, 03:17 AM
mr fantastic
Quote:

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.