Consider polynomials in X of degree up, but not including N. The powers 1,X,...,X^(n-1) form a basis. Then the coefficients of the polynomial can be put in a column vector. If D is the derivative operator, DX^n = nX^(n-1), so the derivative matrix can be expressed as a sparse matrix with D_(n,n+1) = n. Visually, it's a matrix with the integers 1,2,...,n-1 on the super-diagonal.
You can also see that this is a nilpotent matrix for finite N, since repeated multiplication sends the entries further up into the upper right corner.
You can extend this to the infinite case for formal power series in X, too, where you don't worry about convergence.
> Google's PageRank is a solution of a matrix equation, what does that matrix represent?
Isn't it just the adjacency matrix of a big graph?
Anyway, I agree with you. Matrices and linear algebra is a really good inspiration for higher level concepts like vector spaces and Hilbert spaces and so on. That's where the real power lies. But even in such general domains, matrices are often used to do concrete computations on them, because we have a lot of tools for matrices.
You don't even have to go 3D, just starting with the points of a rectangle in 2D and asking, "how do you put the edge points of this rectangle 10px to the left, rotate them 45° and stretch them 200% vertically?" and you've applied a matrix. Even if you're not using the fancy brackets, you're using a matrix, and understanding it.