If αandβ are transformations on a set S, prove that α−1∘β−1 and β−1∘α−1 are transformations. Which is the inverse of α∘β ? Prove your answer.

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Bentley Leach · Accepted Answer

Step 1  Linear transformation are represented as the matrix .  (AB)−1(B−1A−1)=I  If A and B are two matrix then (AB)−1=B−1A−1or,(B−1A−1)(AB)−1=I  So,(α∘β)−1=β−1∘α−1  Let us prove the (α∘β)−1=β−1∘α−1  Let us consider s∈S  Show that  (α∘β)−1∘(β−1∘α−1)(s)=Is(s)  (β−1∘α−1)∘(α∘β)−1(s)=Is(s)  (α∘β)−1∘(β−1∘α−1)(s)=αβ(β−1α−1(s))  =α(Is(α−1(s)))  =α(α−1(s))  =Is(s)  Step 2  (β−1∘α−1)∘(α∘β)−{1}(s)=β−1(α−1α(β(s)))  =β(Is(β−1(s)))  =β(β−1(s))  =Is(s)  So,(α∘β)−1=β−1∘α−1  Hence proved

If \alpha and \beta are transformations on a set S, prove that \alpha^{

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