Condition for Composition of Linear Real Functions to be Commutative

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Theorem

Let $a, b, c, d \in \R$ be real numbers.

Let $\theta_{a, b}: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$

Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$.

Then:

$\theta_{c, d} \circ \theta_{a, b} = \theta_{a, b} \circ \theta_{c, d}$

if and only if:

$b c + d = a d + b$


Proof

\(\displaystyle \map {\theta_{c, d} \circ \theta_{a, b} } x\) \(=\) \(\displaystyle \map {\theta_{a, b} \circ \theta_{c, d} } x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \theta_{a c, b c + d}\) \(=\) \(\displaystyle \theta_{c a, a d + b}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle b c + d\) \(=\) \(\displaystyle a d + b\)

$\blacksquare$


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