Linear transformations

Why this matters

So far a matrix has been a table of coefficients. There is a more powerful way to see it: a matrix is a function that moves vectors. It can rotate, stretch, shearA transformation that slides one direction in proportion to the other, leaning a square while keeping its area. or flatten space. Computer graphics and game engines rotate every vertex with a matrix, robotics chains rotations to place a gripper, materials scientists describe strain with a matrix, and data scientists reduce dimensions by projecting data. Once you see matrices as transformations, eigenvalues (next module) make sense.

The core idea

A linear transformationA function on vectors with T(u + v) = T(u) + T(v) and T(c·v) = c·T(v); it keeps grid lines straight and the origin fixed. T takes a vector in and gives a vector out, with two rules: T(u + v) = T(u) + T(v) and T(c·v) = c·T(v). Grid lines stay straight, parallel and evenly spaced, and the origin stays put. Every linear transformation of 2D space is multiplication by a 2 × 2 matrix, and the matrix is easy to read:

The columns of the matrix are where the basis vectorsThe vectors i = (1, 0) and j = (0, 1); a matrix's columns show where they land. land.

The basis vectors are i = (1, 0) and j = (0, 1). If a transformation sends i to (a, c) and j to (b, d), its matrix is

| a  b |
| c  d |

and any vector (x, y) = x·i + y·j lands at x·(a, c) + y·(b, d).

A gallery of transformations

Worked example: rotate the point (2, 1) by 90°.

| 0  −1 | | 2 |   | 0·2 − 1·1 |   | −1 |
| 1   0 | | 1 | = | 1·2 + 0·1 | = |  2 |

Composition is multiplication

Applying B first and then A is the single transformation AB. Order matters: rotating then shearing is not the same as shearing then rotating, which is exactly why matrix multiplication is not commutative. Two rotations combine as you would hope: rotation(α) · rotation(β) = rotation(α + β).

The determinant as area scale

The determinantFor a transformation, the factor by which it scales area; negative means orientation flips and 0 means space is flattened. of a 2 × 2 matrix tells you how much the transformation scales area. The unit square (area 1) becomes a parallelogram of area |det|:

• Scaling by 2 and 3: det = 6, areas grow six-fold.
• Any rotation: det = 1, areas unchanged.
• A shear: det = 1, the square leans but keeps its area.
• A projection: det = 0, space is flattened onto a line, which is why it has no inverse.
• A negative determinant means the transformation flips orientation (like a mirror).

In code

import numpy as np

def rotation(theta):
    c, s = np.cos(theta), np.sin(theta)
    return np.array([[c, -s], [s, c]])

R = rotation(np.pi / 2)
print(R @ np.array([2.0, 1.0]))    # about [-1, 2]
print(np.linalg.det(R))            # 1.0

To transform many points at once, stack them as columns of a matrix P and compute R @ P.

Common mistakes

• Putting the images in rows instead of columns. The column convention is what makes A @ x work.
• Angles in degrees. np.cos expects radians; convert with np.radians.
• Reading AB as "A first". In AB applied to x, B acts first.
• Forgetting translation is not linear. Moving every point by a fixed offset moves the origin, so it is not a 2 × 2 matrix. Graphics uses 3 × 3 "homogeneous" matrices to include it.

Quick recap

• A matrix is a linear transformation: its columns are the images of the basis vectors.
• Rotations, scalings, shears, reflections and projections all have simple matrices.
• Composing transformations multiplies their matrices, and order matters.
• The determinant is the area scale factor; 0 means space is flattened.

Try it

In the Do tab, build a rotation matrix and test that it behaves like a rotation.