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?

You ask

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?

Other answers and comments address the connection between linear transformations and systems of equations. You can turn questions about systems of equations into questions about their matrices of coefficients and then about linear transformations, and vice versa. The sum and product operations on matrices correspond to addition and composition of linear transformations. That back and forth illuminates both areas.
Abstractly, a matrix is just a rectangular array of numbers. That can be a useful abstraction in contexts other than systems of equations. It's the underlying abstraction in a spreadsheet. For example, the percentages of urban, suburban and rural areas by state will be a $50 \times 3$ matrix. In that context matrix addition is not likely to be useful.
