Can we think of matrices as functions of sorts?

Question

Can we think of matrices as surjective functions where; $i,j$ (the indices) represent the inputs and $z$ (the element at that particular position)? Further, ($i,j,z$) as an ordered triplet...

The reason why I ask, is because, then that could possibly allow us to think of matrix multiplication as a composition of sorts, right?

I think that this approach works, but I'm new to the matrices, so I wanted to check here. I've heard of linear maps, but I don't think that they're what I'm aiming for. Are they?

This is in fact essentially how you prove that the matrices are exactly the linear maps for finite dimensional vector spaces. There might even be an exercise like that in your book. — Malady, Feb 10 '24 at 17:16
for it to be surjective (or not) you have to tell us what kind of elements you place in it. — Phicar, Feb 10 '24 at 17:17
Understood. Should I, um, delete this or is it fine to leave it up here? — idk, Feb 10 '24 at 17:18
Related and possibly interesting: correspondence between $2×2$ matrices and linear fractional transformations — MJD, Feb 10 '24 at 17:35

score 7 · Accepted Answer · answered Feb 10 '24 at 17:27

You can indeed think of a matrix as a function sending the indices to the corresponding entry. Technically speaking, this is exactly the definition of a matrix. The notation $A = (a_{ij})_{1 \leq i \leq n, 1 \leq j \leq m}$ for an $n \times m$-matrix shows this. $A$ is the function $(i, j) \mapsto a_{ij}$.

There is also another way to interpret matrices as functions. Namely, you can interpret a matrix $A$ as the unique linear map $f_A$ sending the $i$-th standard vector to the $i$-th column of $A$. This is the linear map $f_A \colon x \mapsto Ax$ using matrix-vector-multiplication. Using this interpretation, multiplication of matrices corresponds exactly to composition of functions, as can be seen by a computation: $$ (f_A \circ f_B)(x) = f_A(f_B(x)) = A(Bx) = (AB)x = f_{AB}(x) $$ This interpretation of matrices is a very important point in linear algebra. One proves that after a choice of bases all linear maps between finite-dimensional vector spaces can be described by a matrix in this way.

Can we think of matrices as functions of sorts?

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