Exercise 9

Show that if the set $\{\hskip 1.00006pt\vec{u},\vec{v},\vec{w}\hskip 1.00006pt\}$ is linearly independent then so is the set $\{\hskip 1.00006pt\vec{u},\vec{u}+\vec{v},\vec{u}+\vec{v}+\vec{w}\hskip 1.00006pt\}$.

What is the relationship between the linear independence or dependence of $\{\hskip 1.00006pt\vec{u},\vec{v},\vec{w}\hskip 1.00006pt\}$ and the independence or dependence of $\{\hskip 1.00006pt\vec{u}-\vec{v},\vec{v}-\vec{w},\vec{w}-\vec{u}\hskip 1.00006pt\}$?