Linear transformations and matrices

Multiply M = [[2, 5], [1, 3]] by the basis vector ⟨1, 0⟩ and the row-by-column machinery picks out exactly the first column: ⟨2, 1⟩. Multiply by ⟨0, 1⟩ and out comes the second column, ⟨5, 3⟩. That is the reading rule for every 2 × 2 matrix: column 1 is the image of (1, 0), column 2 is the image of (0, 1). Rows are the dot-product workers; columns are the destinations.

Because every point is a combination of the two basis vectors, those two destinations determine everything. The point (1, 1) = (1, 0) + (0, 1) lands at column 1 + column 2: under [[2, 1], [0, 3]] it goes to ⟨2, 0⟩ + ⟨1, 3⟩ = ⟨3, 3⟩, which is exactly what the product [[2, 1], [0, 3]] · ⟨1, 1⟩ computes.

The column rule runs backward, which is how matrices get built. If a transformation sends (1, 0) to (3, 1) and (0, 1) to (−2, 4), its matrix is those images stacked as columns: [[3, −2], [1, 4]]. Stacking them as rows instead transposes the machine into a different transformation.

The famous machines are all column-built this way:

1. Rotation by θ counterclockwise: (1, 0) swings to (cos θ, sin θ) and (0, 1) to (−sin θ, cos θ), giving [[cos θ, −sin θ], [sin θ, cos θ]]. For 90°: [[0, −1], [1, 0]].
2. Reflection across the x-axis: (1, 0) stays, (0, 1) flips to (0, −1): [[1, 0], [0, −1]].
3. Dilation by k about the origin: both basis vectors stretch by k: [[k, 0], [0, k]].

A matrix transformation acts on every point of the plane simultaneously: the plane itself is stretched, rotated, sheared, or flipped, carrying all its points, lines, and regions along. It does not "move the point to the matrix's entries," and it does not single out one point while leaving the rest alone. Lines through the origin map to lines through the origin, and the origin never moves: M · ⟨0, 0⟩ = ⟨0, 0⟩ for every M.

Area rides along too, scaled everywhere by the same factor |det M|, with a negative determinant recording that the plane got flipped over (orientation reversed) in the process. Under [[3, −2], [1, 4]], det = 3·4 − (−2)·1 = 14: every region's area is multiplied by 14, orientation preserved.

Transformations chain: apply A, then apply B to the result. The net effect is a single matrix, the product BA, written right to left because the input vector meets A first: B(A·v) = (BA)·v.

Order matters exactly as much as it did for matrix products, because they are the same thing. Rotate 90° counterclockwise (R = [[0, −1], [1, 0]]) and then reflect across the x-axis (F = [[1, 0], [0, −1]]): the net machine is FR = [[0, −1], [−1, 0]]. In the other order the net machine is RF = [[0, 1], [1, 0]], a different transformation. Track one point to feel it: (1, 0) rotates to (0, 1) then reflects to (0, −1); reflected first it stays at (1, 0), then rotates to (0, 1). Same two machines, different journeys.