Condition for Composition of Linear Real Functions to be Commutative
Jump to navigation
Jump to search
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:
- $\theta_{c, d} \circ \theta_{a, b} = \theta_{a, b} \circ \theta_{c, d}$
- $b c + d = a d + b$
Proof
\(\ds \map {\theta_{c, d} \circ \theta_{a, b} } x\) | \(=\) | \(\ds \map {\theta_{a, b} \circ \theta_{c, d} } x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \theta_{a c, b c + d}\) | \(=\) | \(\ds \theta_{c a, a d + b}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds b c + d\) | \(=\) | \(\ds a d + b\) |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $3$. Mappings: Exercise $5$