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[parent] examples of non-commutative operations (Example)

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

$\displaystyle A=\begin{pmatrix} 1 & 1\ 0&1 \end{pmatrix},\qquad B=\begin{pmatrix} 0 & 1\ 0 & 1 \end{pmatrix}$

If we compute $ AB$ we get

$\displaystyle AB=\begin{pmatrix} 0 & 2 \ 0 & 1 \end{pmatrix}$
but if we compute $ BA$ we have
$\displaystyle 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

$\displaystyle f(x) = x^2,\qquad g(x) = 2x. $

We see that

$\displaystyle (f\circ g)(x) = f(g(x)) = (2x)^2 = 4x^2, $
but
$\displaystyle (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.



"examples of non-commutative operations" is owned by yark. [ full author list (3) | owner history (2) ]
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Cross-references: real functions, composition, function, numbers, operations, commutative, matrix product, matrices, integer, matrix multiplication, non-commutative operation

This is version 7 of examples of non-commutative operations, born on 2005-02-17, modified 2006-02-17.
Object id is 6768, canonical name is ExampleOfCommutative.
Accessed 2191 times total.

Classification:
AMS MSC20-00 (Group theory and generalizations :: General reference works )

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