Question about doing $f$ on basis vectors.

We have $f:V_1\to V_2$ linear map and let's $(e_1,..,e_n)$ be basis for $V_1$ and $(c_1,..,c_n)$ be different basis for $V_1$ $(e_1,..,e_n)=(c_1,..,c_n)\Gamma_1$ where $\Gamma_1$ is change of basis matrix from $(c_1,..,c_n)$ to $(e_1,..,e_n)$.

Then book says that from here follows that $(f(e_1),f(e_2),..,f(e_n))$=$(f(c_1),f(c_2),..,f(c_n))$$\Gamma_1$

Can you explain how author got this?And one more thing $\Gamma_1$ defined with that formula is change of basis matrix from $(c_1,..,c_n)$ to $(e_1,..,e_n)$ not from $(e_1,..,e_n)$ to $(c_1,..,c_n)$.Am I right?

Let's choose to write in the basis $c_1,...,c_n$. Then $e_i = \sum_j (\Gamma_1)_{ji} c_j$. Since $f$ is a linear map, $f(e_i) = \sum_j(\Gamma_1)_{ji}f(c_j)$. This is exactly the statement that $(f(e_1),...,f(e_n)) = (f(c_1),...,f(c_n))\Gamma_1$. And yes, $\Gamma_1$ is the change of basis $c$ to $e$, since (as we have seen above), it allows one to write the vectors $e_i$ in terms of the $c$ basis.