I want to understand all possible homomorphisms $$\alpha: \mathbb{Z}^a \to \mathbb{Z}^b$$ as well as understand what a matrix representation for an arbitrary one of these homomorphisms would look like. Furthermore, under what conditions does a homomorphism have a matrix representation?
To begin, let $(e_1,e_2,......,e_a)$ be a bases for $\mathbb{Z}^a$ and $(f_1,f_2,......,f_b)$ be a basis for $\mathbb{Z}^b$.
And eh, yeah, can someone give me some insights into my inquiries? Thanks!
Just write $\alpha(e_j) = \sum_{i=1}^b m_{ij} f_i$ for each $j=1,\dots,a$.
Then, the matrix $M=(m_{ij})\in \mathbb Z^{b \times a}$ describes $\alpha$ in the sense that $[\alpha(v)]_f = M[v]_e$.
This works exactly as with linear maps between vector spaces.