Comment by bmacho
A matrix is just a grid of numbers.
A lot of areas use use grid of numbers. And matrix theory actually incorporates every area that uses grids of numbers, and every rule in those areas.
For example the simplest difficult thing in matrix theory, matrix multiplication is an example for this IMO. It looks really weird in the context of grid of numbers, and its properties seem incidental, and the proofs are complicated. But matrix multiplication is really simple and natural in the context of linear transformations between vector spaces.
This is the most important part.
"...linear transformations between vector spaces."
When you understand what that implies you can start reasoning about it visually.
The 3 simplest (that you can find in blender or any other 3d program, or even partly in 2d programs).
Translation (moving something left,right,up,down,in,out).
Rotation (turn something 2 degrees, 90 degrees, 180 degrees, 360 degrees back to the same heading)
Scaling (make something larger, smaller, etc)
(And a few more that doesn't help right now)
The 2 first can be visualized simply in 2d, just take a paper/book/etc. Move it left-right, up down, rotate it.. the book in the original position and rotation compared to the new position and rotation can be described as a vector space transformation, why?
Because you can look at it in 2 ways, either the book moved from your vantage point, or you follow the book looking at it the same way and the world around the book moved.
In both cases, something moved from one space (point of reference) to another "space".
The thing that defines the space is a "basis vector", basically it says what is "up", what is "left" and was in "in" in the way we move from one space to another.
Think of it as, you have a piece card on a paper. Draw an line/axis along the bottom edge as the X count, then draw on the left side upwards the Y count. In the X,Y space (from space) you count the X and Y steps of various feature points.
Now draw the "to space" as another X axis and another Y axis (could be rotated, could be scaled, could just be moved) and take the counts in steps and put them inside the "to space" measured in equal units as they were in the from space.
Once the feature points are replicated in the "to space" you should have the same image as before, just within the new space.
This is the essence of a so called linear(equal number steps) transform (moved somewhere else), and also exactly what multiplying a set of vectors by a matrix achieves (simplified, in this context, the matrix really is mostly a representation of a number of above mentioned basis vectors that defines the X, Y,etc of the movement).