I am self teaching module theory and in the section on modules over PID, I have ran into the following problem. Show that the cardinality of M= $\mathbb{Z}^n/T(\mathbb{Z}^n)$ equals the absolute value of the determinant of the matrix representing the homomorphism T. Note: Assuming determinant of the matrix is not 0.
I am thinking let $e_1,...e_n$ be the standard basis for $Z^n$. Then $T(e_1)= c_11 e_1 + ... c_1n e_n$, etc.. so if I find the invariant factors of M by applying column and row operations to the matrix that represents this transformation until its a diagonal matrix, then I will have 1's and invariant factors on the diagonal, and hence M will be isomorphic to $Z^n/(a_1) \bigoplus ... \bigoplus Z^n/(a_n)$ where the $a_i$ are the invariant factors and hence the cardinality will just be the product of the invariant factors, which I guess should be the absolute value of the determinant representing the matrix? I am not sure if this approach would work or not. Could someone help me?
