How are matrices used to represent linear transformations?

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modern algebra A simple algebraic curve, showing the graph of the algebraic equation y² = x³ + 1.

In linear algebra, linear transformations are functions that map one vector to another while preserving vector addition and scalar multiplication. To perform computations with these transformations, they are represented as matrices—rectangular arrangements of scalars.1

The matrix representation of a transformation is determined using coordinates. In two dimensions, a transformation $T$ is defined by its effect on two vectors with different directions. Because the transformed vectors $T (v)$ and $T (w)$ each require two coordinates, these four values are arranged into a $2 \times 2$ matrix. This logic extends to higher dimensions; for instance, a transformation in three dimensions is represented by a $3 \times 3$ matrix.1

Matrices are powerful tools because their algebraic properties mirror the behavior of the transformations they represent. Specifically, the product of two matrices corresponds to the composition of two linear transformations. Because the order of operations matters in transformations, matrix multiplication is generally non-commutative, meaning the product $S T$ is rarely equal to $T S$ .1