The deepest idea in the subject is that a matrix is not really a grid of numbers at all: it is a motion of space. Multiplying vectors by a fixed matrix transforms the whole plane, and understanding that transformation is understanding the matrix.
What "linear" means geometrically
A transformation is linear when it keeps space straight and evenly ruled: gridlines stay straight, parallel, and equally spaced, and the origin does not move. No bending, no curving, no shifting the centre. Rotating the plane about the origin is linear. Stretching it is linear. Skewing it into a slanted grid is linear. Bending it into a curve, or sliding everything sideways, is not.
A matrix is a transformation
Here is the key fact: a linear transformation is completely determined by what it does to a couple of basic arrows, the unit vectors along the axes. Wherever those land, everything else follows, because every vector is just a linear combination of them. And where they land is exactly what the columns of the matrix record. That is why the matrix-vector product combines the columns: the columns are the images of the axes.
A gallery
- A rotation turns every vector about the origin by a fixed angle.
- A scaling stretches or shrinks along the axes.
- A shear slides one direction past another, tipping squares into parallelograms.
- A projection flattens space onto a line, losing a dimension.
Each is a linear transformation, and each is captured by a single matrix.
Composition
Applying one transformation and then another gives a new linear transformation, and the matrix for the combined motion is the product of the two matrices. This is what matrix multiplication really means: it is the algebra of doing one thing after another to space. Seen this way, an equation like asks a geometric question (which starting vector does the motion carry to ?), and the rest of the course answers it.