I understand that the determinant of a matrix can be written in terms of a fermionic path integral. The expression is:
$$Z = \int D\bar{\psi}D\psi e^{-\iint d^4x' d^4x \bar{\psi}(x')B(x',x)\psi(x)}\tag{1}$$
The proof proceeds by rewriting the complex Grassmann fields in terms of basis functions:
$$\psi(x) = \sum_n c_n\chi_n(x)\qquad \text{and}\qquad \bar{\psi}(x) = \sum_n \bar{c_n}\chi_n^\dagger(x)\tag{2}$$
This induces a change in the measure in the form: $$D\bar{\psi}D\psi = \prod_nd\bar{c_n}dc_n.\tag{3}$$
I am failing to prove that the measure indeed changes in the above manner under the stated transformation. An explicit calculation would be much appreciated.