What's the relationship between linear transformations and systems of equations? I began watching Gilbert Strang's lectures on Linear Algebra and soon realized that I lacked an intuitive understanding of matrices, especially as to why certain operations (e.g. matrix multiplication) are defined the way they are. Someone suggested to me 3Blue1Brown's video series (https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) and it has helped immensely. However, it seems to me that they present matrices in completely different ways: 3Blue1Brown explains that they represent linear transformations, while Strang depicts matrices as systems of linear equations. What's the connection between these two different ideas?
Furthermore, I understand why operations on matrices are defined the way they are when we think of them as linear maps, but this intuition breaks when matrices are thought of in different ways. Since matrices are used to represent all sorts of things (linear transformations, systems of equations, data, etc.), how come operations that are seemingly defined for use with linear maps the same across all these different contexts?

First of all, simply by using matrix multiplication, you can rewrite $$Ax$$
into some equations
$$a_{11}x_1+\cdots+a_{1n}x_n$$
$$\cdots$$
$$a_{m1}x_1+\cdots+a_{mn}x_n$$
This means, that a matrix does nothing more than being a representation of the coefficients of these linear equations, which is more convenient to write down.
If you think of it in a more abstract way, you could even just write down
$$a_{11},\ldots, a_{1n},a_{21},\ldots,a_{2n},\ldots,a_{m1},\ldots,a_{mn}$$
into one vector or text file, where $a_{ij}$ are just some variables for anything. The total amount of variables is $m\cdot n$.
Now, if you have given a matrix $A$ and you vary $x$ you can think of $A$ as a linear transformation of $x$. For example the matrix
$$\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}$$
will map the plane $\mathbb{R}^2$ onto a plane which has a $45°$ angle with the $x_1$-axis. So in this case it might be better to think of $A$ as a map, which transforms $x$ in a linear way.
However, if you want to think of systems of linear equation with a given $b$, you can rewrite $$Ax=b$$
into some equations
$$a_{11}x_1+\cdots+a_{1n}x_n=b_1$$
$$\cdots$$
$$a_{m1}x_1+\cdots+a_{mn}x_n=b_m$$
Now, you are more likely to have an interest in not transforming the $x$, but finding an $x$ that is a solution to the equations. In this case you are more likely to think of a system of equations, which needs a solution.
