Exercise 15

Let $\langle\vec{\beta}_{1},\vec{\beta}_{2},\vec{\beta}_{3}\rangle$ be a basis for a vector space.

• Show that $\langle c_{1}\vec{\beta}_{1},c_{2}\vec{\beta}_{2},c_{3}\vec{\beta}_{3}\rangle$ is a basis when $c_{1},c_{2},c_{3}\neq 0$. What happens when at least one $c_{i}$ is $0$?

• Prove that $\langle\vec{\alpha}_{1},\vec{\alpha}_{2},\vec{\alpha}_{3}\rangle$ is a basis where $\vec{\alpha}_{i}=\vec{\beta}_{1}+\vec{\beta}_{i}$.