1. ## Prove linearly independent

Prove the statement:

If u, v, and w are linearly independent, then u+v, v, and w are also linearly independent.

2. Originally Posted by ssl000
Prove the statement:

If u, v, and w are linearly independent, then u+v, v, and w are also linearly independent.
Given: $\alpha \vec{u} + \beta \vec{v} + \gamma \vec{w} = \vec{0}$ only when $\alpha = \beta = \gamma = 0$ are equal to zero.

Now re-write $\alpha \vec{u} + \beta \vec{v} + \gamma \vec{w}$ as $\alpha (\vec{u} + \vec{v})+ \delta \vec{v} + \gamma \vec{w} = \vec{0}$ ....

3. Originally Posted by mr fantastic
Given: $\alpha \vec{u} + \beta \vec{v} + \gamma \vec{w} = \vec{0}$ where not all of the $\alpha, \beta, \gamma$ are equal to zero.

This can be re-written as $\alpha (\vec{u} + \vec{v})+ \delta \vec{v} + \gamma \vec{w} = \vec{0}$ where $\delta = \beta - \alpha$ ....
Wouldn't that prove that its linearly dependent, since $\delta = \beta - \alpha$?

4. Originally Posted by ssl000
Wouldn't that prove that its linearly dependent, since $\delta = \beta - \alpha$?