A matrix is a verb
The idea that unlocks linear algebra: a matrix is not really a grid of numbers, it is a function that transforms space. Feed it a vector, it hands back a moved vector. The whole plane gets picked up and reshaped, and the matrix records exactly how.
What makes a transformation linear is a strict promise: grid lines stay straight and evenly spaced, and the origin stays put. No curving, no bending. Under that promise, the entire transformation is pinned down by what it does to just two vectors.
The faint grid is space before; the green grid is space after. Drag the sliders — they ARE the matrix. Watch î (the first column) and ĵ (the second column) land exactly where the matrix’s columns say they should.
The columns are where the basis vectors land
Here is the punchline that makes matrices readable at a glance. Start with the two basis vectors: î = [1, 0] (one step east) and ĵ = [0, 1] (one step north). A linear transformation is completely determined by where these two go.
And where do they go? The columns of the matrix are exactly the landing spots. The first column is the new î; the second column is the new ĵ. That is the entire meaning of a 2×2 matrix. Once you know where î and ĵ land, every other vector follows, because every vector is a linear combination of î and ĵ — and linearity preserves those combinations.
Read any 2×2 matrix instantly
The matrix [[a, b], [c, d]] sends î to the column [a, c] and ĵ to the column [b, d]. That is why a matrix "is" a transformation: its columns tell you the new grid, and the new grid tells you everything.
The famous transformations
A handful of matrices come up constantly. Each one is just a statement about where î and ĵ go:
| Transformation | Matrix (columns = new î, ĵ) | What it does |
|---|---|---|
| Identity | [[1,0],[0,1]] | nothing — the do-nothing map |
| Scale ×2 | [[2,0],[0,2]] | everything twice as far from origin |
| Rotate 90° CCW | [[0,-1],[1,0]] | î → up, ĵ → left |
| Reflect in x-axis | [[1,0],[0,-1]] | flips top and bottom |
| Shear | [[1,1],[0,1]] | slides the top sideways, base fixed |
| Collapse to a line | [[1,2],[2,4]] | squashes the plane onto one line |
Composition is matrix multiplication
Apply one transformation, then another, and the combined effect is a single transformation — whose matrix is the product of the two. This is the real reason matrix multiplication is defined the strange way it is: it is built so that the product matrix does exactly "first this map, then that one".
And it is why order matters. The product BA means "do A first, then B" (right to left, like function composition). Rotate-then-shear and shear-then-rotate reshape space differently, so BA ≠ AB — the non-commutativity from the last lesson is just this geometric fact.
A worked example
Where does the matrix R = [[0, -1], [1, 0]] send the vector [2, 3], and what transformation is R?
- Read the columns: î = [1,0] lands at [0, 1]; ĵ = [0,1] lands at [-1, 0].The columns of R are the images of the basis vectors.
- î went from east to north, ĵ from north to west — a 90° anticlockwise rotation.Rotating the whole plane 90° CCW sends east→north and north→west.
- Apply to [2, 3]: 2·[0,1] + 3·[-1,0] = [0,2] + [-3,0] = [-3, 2].Every vector is carried by the same combination of the new basis vectors.
R rotates the plane 90° anticlockwise, sending [2, 3] to [-3, 2]. Notice the length is unchanged (√13 both times) — rotations preserve distance.