# Is value in the matrix notation

Given an $$m\times n$$ matrix $$A=\begin{bmatrix} a_{1,1}&...&a_{1,n} \\ \vdots&\ddots&\vdots \\ a_{m,1}&...&a_{m,n} \\ \end{bmatrix}$$, say that I wanted to describe a set generator $$S(A)$$ which only consists of every individual entry in A, using set builder notation. I want to write something like this...

$$S(A) = \left\{s\in\mathbb{R}\ :\ s\in A\right\}$$

But I know that '$$s\in A$$' is incorrect bc '$$\in$$' should only ever be used wrt sets; is there a common convention (perhaps another symbol) that I could use?

What is a matrix, exactly?

A good formal definition of a matrix is a function

$$A : [m] \times [n] \rightarrow \mathbb{R}$$

where $$[m] = \{1,2,\cdots,m\}$$ and $$[n] = \{1,2,\cdots,n\}$$. The codomain could be any set, but let's use $$\mathbb{R}$$ for sake of example.

Thus, when you see an entry $$a_{i,j}$$, you can essentially think of it as the output $$A(i,j) \in \mathbb{R}$$. In this case, you can easily see that the notation you're looking for is simply the range $$ran(A)$$of $$A$$:

$$ran(A) = \{A(i,j) | (i,j) \in [m]\times [n]\} = \{a_{i,j} | (i,j) \in [m] \times [n]\}$$

• I strongly recommend not using $\operatorname{ran}(A)$ for the set of entries of $A$ because that notation is often used for the range of $A$ considered as a linear map $\mathbb R^n \to \mathbb R^m$. You can use the other two notations on the last line, though. Jan 16, 2019 at 7:35