How can I show that there exists a Linear Transformation $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ s.t. $f(l_i)=m_i$ for two sets of three distinct lines ${l_1,l_2,l_3}$ and ${m_1,m_2,m_3}$ each of which passes through the origin.
I was trying to do this with a matrix equation where a $2 \times 2$ matrix represented the linear transformation and each line was represented by a vector. This gave rise to a system of six equations and six unknowns but I don't know how to guarantee a solution exists.