ExampleOfCommutative 2005-02-17 20:56:00 2006-02-17 07:56:40 Example commutative \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\Z}{\mathbb{Z}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} A standard example of a non-commutative operation is matrix multiplication. Consider the following two integer matrices: \[ A=\begin{pmatrix} 1 & 1\\ 0&1 \end{pmatrix},\qquad B=\begin{pmatrix} 0 & 1\\ 0 & 1 \end{pmatrix} \] If we compute $AB$ we get \[ AB=\begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} \] but if we compute $BA$ we have \[ BA=\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}. \] Since $AB\neq BA$ we conclude that matrix product is not commutative. Operations do not necessarily have to operate on numbers. Another classic example is function composition. Let $f$ and $g$ be real functions given by \[ f(x) = x^2,\qquad g(x) = 2x. \] We see that \[ (f\circ g)(x) = f(g(x)) = (2x)^2 = 4x^2, \] but \[ (g \circ f )(x) = g(f(x)) = 2(x^2) = 2x^2. \] Since we got different functions, we conclude that function composition is not commutative.