Matrices as functions

A 2×2 matrix $A$ defines a function on the plane: input a vector $v$, output the matrix-vector product $Av$. The mechanics: each row of the matrix dots with the input to produce one coordinate of the output. For $A = [[2, 1], [0, 3]]$ and $v = \langle 4, 5\rangle$:

1. First output coordinate: row one dot v: $2 \cdot 4 + 1 \cdot 5 = 13$.
2. Second output coordinate: row two dot v: $0 \cdot 4 + 3 \cdot 5 = 15$.

So $A\langle 4, 5\rangle = \langle 13, 15\rangle$. Every coordinate of the output uses every coordinate of the input; nothing is multiplied "slot by slot." The matrix transforms the whole plane at once: every vector has an assigned output, exactly like $f(x) = 2x + 1$ assigns an output to every number.

Run the two simplest inputs through the machine. $A\langle 1, 0\rangle$ picks out the first column of A, and $A\langle 0, 1\rangle$ picks out the second column. That is the decoder ring for every matrix map: the columns are the images of the basis vectors.

It works in both directions. Reading: $[[3, -1], [2, 4]]$ sends $(1, 0)$ to $(3, 2)$ and $(0, 1)$ to $(-1, 4)$; sketch those two images and you can see what the map does to the whole grid. Building: if a transformation sends $(1, 0) \to (0, 1)$ and $(0, 1) \to (-1, 0)$ (a quarter-turn counterclockwise), install those images as columns: $[[0, -1], [1, 0]]$. Two basis images determine the entire map, because every other vector is a combination of the basis and the map respects combinations.

Doing one matrix map after another is itself a matrix map, and its matrix is the product. The order trap: applying A first and then B computes $B(Av)$, which is the product $BA$, the later map written on the left, matching function notation $f(g(x))$ where the inside function acts first.

And order genuinely matters. Take the shear $S = [[1, 1], [0, 1]]$ and the quarter-turn $R = [[0, -1], [1, 0]]$, applied to $\langle 0, 1\rangle$. Shear first: $S\langle 0,1\rangle = \langle 1, 1\rangle$, then $R\langle 1,1\rangle = \langle -1, 1\rangle$. Rotate first: $R\langle 0,1\rangle = \langle -1, 0\rangle$, then $S\langle -1,0\rangle = \langle -1, 0\rangle$. Different destinations, so $RS \ne SR$: matrix multiplication does not commute, because function composition does not.

The identity matrix $I = [[1, 0], [0, 1]]$ is the do-nothing function: $Iv = v$ for every v (its columns say the basis stays put). A matrix $A$ has an inverse $A^{-1}$ exactly when its map can be undone; then $A^{-1}(Av) = v$, so $A^{-1}A = I$. Undo means reverse the process: if A doubles x and triples y, $A^{-1}$ halves x and thirds y. It is not "negate the entries" and not "reciprocal of each entry."

Finally, matrix maps obey a strict contract called linearity: $A(u + v) = Au + Av$ and $A(cv) = c(Av)$. Scale the input, the output scales the same way; add inputs, the outputs add. This is why two basis images determine everything, and it is a genuine restriction: a matrix map can rotate, stretch, shear, and reflect the plane, but it can never shift it, bend it, or move the origin.