Exercise 7

Show that it follows from the prior question that for any six transformations $t_{1},\dots,t_{6}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ there are scalars $c_{1},\dots,c_{6}\in\mathbb{R}$ such that not every $c_{i}$ equals $0$ but $c_{1}t_{1}+\dots+c_{6}t_{6}$ is the zero map. (Hint: the six is slightly misleading.)