Chapter 6 — Linear Transformations
Every matrix is a machine that moves space. Linear transformations explain what that machine does, what it destroys, what it can reach, and how machines are chained together.
Up to now a matrix has been a table of numbers: something to row reduce, multiply, invert or take the determinant of. In this chapter we change our point of view. A matrix $A$ is a machine: feed it a vector $\mathbf{x}$ and it returns a new vector $A\mathbf{x}$. Feed it every point of the plane at once and it moves the whole plane — rotating it, stretching it, shearing it, flipping it over, or flattening it onto a line. Such machines are called linear transformations, and the two rules they obey, $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$, are so simple that they appear almost everywhere.
Every frame of a video game is drawn by multiplying the corners of thousands of triangles by matrices that turn, scale and project a 3-D world onto a flat screen. A robot arm works out where its hand is by chaining one rotation for each joint — and chaining linear maps is exactly matrix multiplication. Photo filters that turn a picture grey or sepia multiply every pixel's colour vector by the same small matrix, and a hospital CT scanner reconstructs an image by "undoing" a linear map from tissue densities to measurements. To use these machines well we need to know what they lose (their kernel), what they can produce (their range), how to describe them with a matrix in any coordinate system, and what they look like geometrically.
| Application | The vectors | The linear map |
|---|---|---|
| Computer graphics | corners of a model in $\mathbb{R}^2$ or $\mathbb{R}^3$ | rotate, scale, shear, project to the screen |
| Robotics | positions of the joints and the hand | one rotation per joint, composed by matrix products |
| Image filters | colour pixels $(R,G,B)$ | greyscale, sepia, colour-space conversion |
| Medical imaging | tissue densities in the pixels | the scanner's measurement map — its kernel is invisible |
- decide whether a map between vector spaces is linear, proving it in general or disproving it with one counterexample;
- find the standard matrix of a linear map $\mathbb{R}^n\to\mathbb{R}^m$ from the images of the standard basis vectors, and compose maps by multiplying matrices in the right order;
- find bases of the kernel and range, decide whether a map is one-to-one or onto from its pivots, and use the rank–nullity theorem to count dimensions;
- recognise isomorphisms and use them to move questions about polynomials and matrices into $\mathbb{R}^n$;
- find the matrix of a linear map relative to any bases, and see how it changes with the basis;
- describe rotations, reflections, shears, projections and scalings of the plane by matrices, read off how they change areas and orientation, and use homogeneous coordinates to include translations.
Sections in this chapter
Linear Transformations
The two linearity rules, non-examples, the standard matrix, maps on polynomials and matrices, and composition as matrix multiplication.
Kernel and Range
What a map loses and what it reaches; one-to-one and onto from pivots; rank–nullity; isomorphisms.
Matrices of Linear Transformations
Describe any linear map by a matrix relative to chosen bases; change of basis and similar matrices $P^{-1}AP$; how a good basis makes the matrix simple.
The Geometry of Linear Maps in the Plane
Rotations, reflections, shears, projections and scalings; composing them; how the determinant measures area and orientation; homogeneous coordinates for translations.