Linear functions define matrix multiplication and make matrices and linear functions effectively interchangeable.
1. Basis
$$
\begin{aligned}
\begin{pmatrix}
x_1 \\
\vdots \\
x_d
\end{pmatrix}
&= x_1
\begin{pmatrix}
1 \\
0 \\
\vdots \\
0
\end{pmatrix}
+ \dots + x_d
\begin{pmatrix}
0 \\
\vdots \\
0 \\
1
\end{pmatrix}\\
&= x_1 \vec{e}_1 + \dots + x_d \vec{e}_d
\end{aligned}
$$
2. Linearity Defines Matrix-Vector Multiplication
$$
\begin{aligned}
f\begin{pmatrix}
x_1 \\
\vdots \\
x_d
\end{pmatrix}
&= f(x_1 \vec{e}_1 + \dots + x_d \vec{e}_d) \\
&= x_1 f(\vec{e}_1) + \dots + x_d f(\vec{e}_d) \qquad \text{(linearity)}\\
&= \begin{pmatrix}
| & & |\\
f(\vec{e}_1) & \dots& f(\vec{e}_d)\\
| & & |
\end{pmatrix}
\begin{pmatrix}
x_1 \\
\vdots \\
x_d
\end{pmatrix}
\end{aligned}
$$
3. Composition Defines Multiplication of Matrices
The property
$$
\begin{aligned}
g\circ f(e_i)
&=
\begin{pmatrix}
| & & |\\
g(\vec{e}_1) & \dots& g(\vec{e}_d)\\
| & & |
\end{pmatrix}
\begin{pmatrix}
| \\
f(\vec{e}_i)\\
|
\end{pmatrix}
\end{aligned}
$$
implies
$$
\begin{pmatrix}
| & & |\\
g\circ f(\vec{e}_1) & \dots& g\circ f(\vec{e}_d)\\
| & & |
\end{pmatrix}
=
\begin{pmatrix}
| & & |\\
g(\vec{e}_1) & \dots& g(\vec{e}_d)\\
| & & |
\end{pmatrix}
\begin{pmatrix}
| & & |\\
f(\vec{e}_1) & \dots& f(\vec{e}_d)\\
| & & |
\end{pmatrix}
$$