Determining linear independence in $\mathbb{R}^3$ Let $\{\xi_k\}_{k=1}^4$ be a set of vectors in $\mathbb{R}^3$. If $\{\xi_1, \xi_2\}$ and $\{\xi_2, \xi_3, \xi_4\}$ are independent sets, and $\xi_1$ belongs to the span of $\{\xi_2, \xi_4\}$. Show that $\{\xi_k\}_{k=1}^3$ is linearly independent. 
Clearly, $\xi_1 = a_1\xi_2 + a_2\xi_4$ where $a_1\neq0$ due to the fact that $\xi_1$ and $\xi_2$ are linearly independent. Hence, $\xi_2 = a_1^{-1}\xi_1-\frac{a_2}{a_1}\xi_4$... How to proceed with the formalization? Because, it is quite clear that $\xi_1$ is linearly independent of $\xi_3$ and we know that $\{\xi_1, \xi_2\}$ is independent as well, so it follows that $\{\xi_k\}_{k=1}^3$ is linearly independent set. Is it rigorous enough?
Thank you! 

hint
Start with a linear combination of vectors $1,2,3$ and set it equal to $0$ (vector). Now write vector $1$ as a linear combination of vectors $2$ and $4$. This gives you a linear combination of vectors $2,3$ and $4$ equal to the zero vector. Now use the independence of $2,3,4$ to find the coefficients.
