Linear transformations
A linear transformation is a function between vector spaces that respects addition and scaling. Matrices are exactly the linear transformations between coordinate spaces.
Definition
A function T from a vector space V to a vector space W is linear if \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\) and \(T(c\mathbf{u}) = cT(\mathbf{u})\) for all vectors and scalars. Equivalently, T preserves linear combinations. Every linear transformation sends the zero vector to zero, so a function such as \(T(x) = x + 1\) is not linear.
Matrix transformations
Every m x n matrix A defines a linear transformation \(T(\mathbf{x}) = A\mathbf{x}\) from \(\mathbb{R}^n\) to \(\mathbb{R}^m\), because matrix multiplication distributes over addition and commutes with scalars. The input space \(\mathbb{R}^n\) is the domain; \(\mathbb{R}^m\) is the codomain; the set of actual outputs, the column space, is the range.
The standard matrix
Conversely, every linear transformation T from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) is a matrix transformation. Its standard matrix has columns \(T(\mathbf{e}_1), \dots, T(\mathbf{e}_n)\), the images of the standard basis vectors. Knowing what T does to a basis determines it everywhere, because any vector is a combination of basis vectors and T preserves combinations.
Geometric examples
In the plane: rotation by \(\theta\) has standard matrix \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\); reflection across the x-axis is \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\); a horizontal shear is \(\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\); projection onto the x-axis is \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\); and scaling by 3 is 3I. Translation is not linear, as it moves the origin.
Linear maps on other spaces
Linearity applies beyond R^n. Differentiation is linear on polynomials or smooth functions: the derivative of f + g is f' + g', and of cf is cf'. Integration from 0 to x is linear. The transpose map on matrices is linear. Expectation in probability is linear, a fact used constantly in statistics.
Composition
Applying S after T gives the composition S o T, which is linear. Its standard matrix is the product of the standard matrices, with the second transformation on the left: [S o T] = [S][T]. This is the fundamental reason matrix multiplication is defined as it is.
Linearity in modelling
Many systems are linear or approximately so for small changes: springs obeying Hooke's law, electrical circuits with resistors, optics, small vibrations, and first-order approximations in calculus, where the derivative is the best linear approximation of a function near a point. Linear models are popular because superposition makes them tractable.
Worked example
Find the standard matrix of T from \(\mathbb{R}^2\) to \(\mathbb{R}^2\) that rotates by 90 degrees counterclockwise and then reflects across the x-axis. Rotation sends \(\mathbf{e}_1\) to (0, 1) and \(\mathbf{e}_2\) to (-1, 0); reflection then gives (0, -1) and (-1, 0). The standard matrix is \(\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}\), which is reflection across the line y = -x, as the product \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) confirms.
Key takeaways
A linear transformation preserves addition and scalar multiplication, hence linear combinations, and sends zero to zero; translations are not linear. Every matrix defines a linear map x -> Ax from R^n to R^m, with range equal to the column space. Every linear map between coordinate spaces has a standard matrix whose columns are the images of the standard basis vectors. Rotations, reflections, shears, projections, and scalings are linear; differentiation, integration, transposition, and expectation are linear on other spaces. Composition corresponds to matrix multiplication. Linear models and linear approximations are central to science and engineering.
Flashcard
Preserves addition and scalar multiplication
Check your understanding
Which function from R to R is linear?