Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Question 7** Consider the set \( \mathcal{L} \) of all linear functions on \( \mathbb{R} \). These are defined as: \[ h(x) = ax + b \] where \( a \in \mathbb{R} \) and \( b \in \mathbb{R} \). Define the addition \( + \) as usual: \[ (f + g)(x) = f(x) + g(x). \] Define the multiplication as the composition of functions: \[ (f \circ g)(x) = f(g(x)). \] Is \( \mathcal{L} \) with these operations \( + \) and \( \circ \) a ring? Explain your answer.
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Transcribed Image Text:**Question 7** Consider the set \( \mathcal{L} \) of all linear functions on \( \mathbb{R} \). These are defined as: \[ h(x) = ax + b \] where \( a \in \mathbb{R} \) and \( b \in \mathbb{R} \). Define the addition \( + \) as usual: \[ (f + g)(x) = f(x) + g(x). \] Define the multiplication as the composition of functions: \[ (f \circ g)(x) = f(g(x)). \] Is \( \mathcal{L} \) with these operations \( + \) and \( \circ \) a ring? Explain your answer.
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