Matrices as linear transformations
A matrix can be read as a machine that moves every vector at once. What does the machine do to space, and when can it be run backward?
Where you are 9 inside
What this area is about
An m \times n matrix A defines a function \mathbf{x} \mapsto A\mathbf{x} from \mathbb{R}^n to \mathbb{R}^m, and every linear transformation between these spaces is one of these. This area reads matrices as such maps: what they reach, what they crush to \mathbf{0}, and when they can be undone.
How these fit together
Definitions. Linear transformations are the maps that respect addition and scaling. The image is the set of outputs, the span of the columns; the kernel is the set of inputs sent to \mathbf{0}.
Procedures. Multiplying a matrix by a vector reads A\mathbf{x} as a combination of the columns. Finding the matrix of a transformation builds A from where each \mathbf{e}_j lands; composing transformations multiplies their matrices.
Results. Rank and nullity: for an m \times n matrix, the dimensions of image and kernel add to n. Invertible transformations need a square matrix whose kernel is \{\mathbf{0}\}, equivalently \det A \ne 0. The invertible matrix theorem gathers a dozen equivalent tests for a square matrix.
A path through it
Matrix times a vector, then linear transformations. After that, finding a transformation’s matrix, composition, the image and the kernel. Rank and nullity needs both image and kernel; invertible transformations needs the kernel. The invertible matrix theorem comes last.
Next come eigenvalues and eigenvectors, the directions a transformation only stretches.