Linear transformations and geometry

So far a matrix has mostly been a table of coefficients in a system of equations. A matrix can also be understood as a machine that transforms vectors: it takes an input vector and produces an output vector, rotating, stretching, reflecting, shearing or projecting it. This point of view explains why matrix multiplication is defined the way it is, what determinants measure, and why eigenvectors (Lesson 9) matter. It is also the working language of computer graphics, robotics, computer-aided design, surveying and the mechanics of deformable materials, where points and directions are constantly rotated, scaled and moved between coordinate systems. This lesson introduces linear transformations, shows how to find the matrix of a transformation from what it does to the basis vectors, catalogs the basic geometric transformations, explains composition and inverses, and shows how homogeneous coordinates bring translations into the same framework.

Linear transformations

A transformation \(T\) from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) assigns to each input vector \(\mathbf{x}\) an output \(T(\mathbf{x})\). It is linear if it respects addition and scaling:

Every matrix defines a linear transformation, \(T(\mathbf{x}) = A\mathbf{x}\), because matrix multiplication has these properties (Lesson 4). Conversely, every linear transformation between spaces of these kinds is multiplication by some matrix. Linear transformations always send the origin to the origin, straight lines to straight lines (or to points), and parallel lines to parallel lines, so they turn grids of squares into grids of parallelograms.

Many physical relationships are linear, at least approximately: the displacement of an elastic structure as a function of the loads, the outputs of a circuit as functions of its inputs, the strain in a material as a function of small deformations. The linear transformation framework applies to all of them.

The matrix of a transformation

A linear transformation is completely determined by what it does to the standard basis vectors \(\mathbf{e}_1, \ldots, \mathbf{e}_n\), because every vector is a combination of them and \(T\) preserves combinations. The matrix of \(T\) has the images of the basis vectors as its columns:

This gives a practical recipe: to find the matrix of a geometric transformation, work out where it sends each unit vector along the axes, and write those images as columns.

A catalog of plane transformations

The determinant has a clear geometric meaning (Lesson 6): its absolute value is the factor by which the transformation scales areas (volumes in three dimensions), and its sign tells whether orientation is preserved (positive) or reversed, as in a reflection (negative). Rotations and shears preserve area; a projection flattens everything onto a line, so its determinant is zero and it cannot be undone.

Projection onto a line

The projection of a vector \(\mathbf{x}\) onto the line through the origin in the direction of a unit vector \(\mathbf{u}\) is \((\mathbf{u}\cdot\mathbf{x})\mathbf{u}\) (Lesson 1). In matrix form this is \(P\mathbf{x}\) with \(P = \mathbf{u}\mathbf{u}^T\), a symmetric matrix satisfying \(P^2 = P\): projecting twice is the same as projecting once. Projections are the basis of least squares (Lesson 10), where a measurement vector is projected onto the set of values a model can produce.

Composition and the order of operations

Applying one transformation after another is called composition. If \(\mathbf{x}\) is transformed first by \(B\) and then by \(A\), the result is \(A(B\mathbf{x}) = (AB)\mathbf{x}\): the matrix of the composition is the product \(AB\), with the first transformation on the right. This is the reason matrix multiplication is defined row-by-column (Lesson 4): it is exactly the rule that makes products represent compositions.

Because matrix multiplication is not commutative, order matters. Rotating and then shearing a figure generally gives a different result from shearing and then rotating it. Rotations about the same axis do commute (\(R(\alpha)R(\beta) = R(\alpha + \beta)\), which proves the angle addition formulas of trigonometry in one line), but three-dimensional rotations about different axes do not, a fact that matters in aircraft attitude, robot joints and camera orientation.

Inverse transformations

A transformation can be undone exactly when its matrix is invertible (determinant non-zero, Lesson 6). The inverse of a rotation by \(\theta\) is the rotation by \(-\theta\), and because \(\cos(-\theta) = \cos\theta\) and \(\sin(-\theta) = -\sin\theta\), this is simply the transpose: \(R^{-1} = R^T\). Matrices with this property, \(Q^TQ = I\), are called orthogonal matrices; their columns are perpendicular unit vectors, and they represent rotations and reflections, which preserve lengths and angles. Their inverses cost nothing to compute, a fact used constantly in graphics and robotics. The inverse of a shear by \(k\) is a shear by \(-k\); the inverse of a scaling by \(s\) is a scaling by \(1/s\); a projection has no inverse.

Rotations in three dimensions

In three dimensions, rotation by \(\theta\) about the \(z\)-axis leaves \(z\) unchanged and rotates \(x\) and \(y\) as in the plane:

with similar matrices \(R_x\) and \(R_y\) for the other axes. Any orientation of a rigid body can be reached by three successive rotations about chosen axes, for example yaw, pitch and roll for an aircraft or a drone. Because these rotations do not commute, the convention for their order must be stated precisely, and mixing conventions is a classic source of errors in navigation and robotics software. Some applications use quaternions, an extension of complex numbers, to represent rotations without the singularities that angle sequences can suffer.

Homogeneous coordinates and translation

A translation, moving every point by a fixed vector \(\mathbf{t}\), is not linear, since it moves the origin. Yet rotations and translations must be combined constantly, for instance to describe the position of a robot gripper relative to the robot's base. The trick is to add an extra coordinate equal to 1, writing the point \((x, y)\) as \((x, y, 1)\). Then a rotation followed by a translation becomes a single matrix multiplication:

Sequences of rotations and translations are then composed by multiplying such matrices, exactly as for linear transformations. Graphics processors perform billions of \(4 \times 4\) homogeneous transformations per second to place objects in a scene and project them onto the screen, and robot controllers chain one such matrix per joint to find where the end of the arm is.

Changing coordinate systems

Engineers constantly move between coordinate systems: a survey grid and a building grid rotated relative to it, a sensor mounted at an angle on a vehicle, the local axes of a beam element and the global axes of the structure that contains it. If the columns of a matrix \(B\) are the new basis vectors written in the old coordinates, then a vector with new coordinates \(\mathbf{c}\) has old coordinates \(\mathbf{x} = B\mathbf{c}\), and conversely \(\mathbf{c} = B^{-1}\mathbf{x}\). When the new basis is orthonormal, as it is for rotated axes, \(B\) is orthogonal and \(B^{-1} = B^T\), so converting coordinates costs only a transpose. A transformation with matrix \(A\) in the old coordinates has matrix \(B^{-1}AB\) in the new ones, which is how the stiffness matrix of a sloping truss member, simple in the member's own axes, is rotated into the global axes of the structure before assembly (Lesson 12).