Differentiation on Polynomials is Linear Operator
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Theorem
Let $\map P \R$ be the vector space of all polynomial functions on the real number line $\R$.
Then the differentiation operator $D$ on $\map P \R$ is a linear operator.
Proof
Let $\map f x, \map g x$ be real functions which are differentiable on $\R$.
Then from Linear Combination of Derivatives:
- $\forall x \in I: \map D {\lambda \map f x + \mu \map g x} = \lambda D \map f x + \mu D \map g x$
It follows from Real Polynomial Function is Differentiable that $\lambda D \map f x + \mu D \map g x$ is differentiable on $\R$.
The result follows from Condition for Linear Transformation.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.6$