Exercise 13

Show that a vector $\vec{\beta}_{0}$ is a linear combination of members of the set $\{\hskip 1.00006pt\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\hskip 1.00006pt\}$ if and only if there is a linear relationship $\vec{0}=c_{0}\vec{\beta}_{0}+\cdots+c_{n}\vec{\beta}_{n}$ where $c_{0}$ is not zero. (Hint. Watch out for the $\vec{\beta}_{0}=\vec{0}$ case.)

Use that to simplify the proof of Lemma 2.5.