$\newcommand{\ones}{\mathbf 1}$

If $f:\mathbf{R}^n \to \mathbf{R}^m$, with $f(x) = Ax$, then

Suppose $A \in \mathbf{R}^{10 \times 10}$ is tridiagonal, i.e., $a_{ij} = 0$ if $|i-j| > 1$, and $y=Ax$. Then $y_4$ does not depend on $x_1$.

Let $x,y \in \mathbf{R}^3$. For $i = 1,2,3$, $y_i$ is the average of $x_1,\ldots,x_i$. Then we have $y = Ax$, with

Let $P \in \mathbf{R}^{m \times n}$ and $q \in \mathbf{R}^n$, where $P_{ij}$ is the price of good $j$ in country $i$, and $q_j$ is the quantity of good $j$ needed to produce some product. Then $(Pq)_i$ is

In the matrix $A\in {\mathbf R}^{m \times n}$, the entries in each row increase in size, moving from left to right. With $y=Ax$, this means

In the matrix $A\in {\mathbf R}^{n \times n}$, the significant entries in the matrix are all near the diagonal. With $y=Ax$, this means

The $j$th column of $AB$ is

The $i$th row of $AB$ is

The $(i,j)$ entry of $AB$ is