Geometry and algebra of linear maps, vectors, and matrices
A matrix is more than a grid of numbers. It is a function that transforms space: feed it a vector x and it hands back a new vector Ax. Across the whole plane it acts as one coherent motion (a rotation, a stretch, a reflection, a shear, a projection) applied to every point at once.
What makes it linear is that it respects the two vector operations: A(x + y) = Ax + Ay and A(cx) = c·Ax. Straight lines stay straight, the origin stays put, and evenly spaced grids map to evenly spaced (possibly slanted) grids.
Here is how to read a matrix by eye: its columns are where the basis vectors land. The first column is the image of [1, 0]; the second column is the image of [0, 1]. Once you know where the two axes go, the whole transformation is fixed, because every other vector is a combination of them.