examples of non-commutative operations


A standard example of a non-commutative operation is matrix multiplicationMathworldPlanetmath. Consider the following two integer matrices:

A=(1101),B=(0101)

If we compute AB we get

AB=(0201)

but if we compute BA we have

BA=(0101).

Since ABBA we conclude that matrix product is not commutativePlanetmathPlanetmathPlanetmath.

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)=x2,g(x)=2x.

We see that

(fg)(x)=f(g(x))=(2x)2=4x2,

but

(gf)(x)=g(f(x))=2(x2)=2x2.

Since we got different functionsMathworldPlanetmath, we conclude that function composition is not commutative.

Title examples of non-commutative operations
Canonical name ExamplesOfNoncommutativeOperations
Date of creation 2013-03-22 15:03:04
Last modified on 2013-03-22 15:03:04
Owner yark (2760)
Last modified by yark (2760)
Numerical id 10
Author yark (2760)
Entry type Example
Classification msc 20-00