PlanetMath: examples of non-commutative operations

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examples of non-commutative operations

(Example)

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

If we compute we get

$\displaystyle AB=\begin{pmatrix} 0 & 2 \ 0 & 1 \end{pmatrix}$

but if we compute

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 and 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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20-00 (Group theory and generalizations :: General reference works )

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