Let $\Gamma$ be a graph and $\{\lambda_1, \dots, \lambda_n\}$ be a maximal set of linearly independent closed paths in $\Gamma$. Let $z\in Z_1(\Gamma)$ be a 1-cycle in $\Gamma$. How can I show that $z$ is a combination of the closed paths $\{\lambda_1,\dots,\lambda_n\}$?
Note: the chain module I'm considering has real coefficients, so it is actually a vector space.