Matrices

A rigorous introduction to matrices as data arrays and representations of linear maps, with shape and entry notation.

Vector-space ideas connect sets, operations, spanning, independence, basis, and dimension. A vector space sits at the centre of a dependency map. Subspace conditions lead to span and independence, which meet at basis and determine dimension; kernel and range connect through a transformation.

Precise definition

An $m\times n$ matrix $A=(a_{ij})$ is a rectangular array with $m$ rows and $n$ columns over a stated field. Its size is part of the object. A matrix can store data, coefficients of a system, or a linear map from $\mathbb F^n$ to $\mathbb F^m$.

Notation and mathematical language

The entry $a_{ij}$ lies in row $i$, column $j$. Columns may be written $A=[a_1\ \cdots\ a_n]$. Square, diagonal, triangular, symmetric ($A^T=A$), identity, and zero matrices describe structural classes, not individual operations.

Conceptual picture

A matrix's meaning comes from context and chosen bases. The same numerical array can represent a data table or a transformation, while the same transformation has different matrices in different bases.

Fully worked example

Interpretation and application

Matrices organize systems, networks, images, transitions, and datasets. Algebraic operations are meaningful only when the representation matches the question; multiplying two data tables merely because dimensions fit may have no sensible interpretation.