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let V be the vector space of all polynomials in the variable x with real the

space V:

        D(P(X))=P'(x) (the derivative of P(x))


prove that both D and M are linear transformations such that DM-MD=I(the identity linear map

on V.

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check that both $M$ and $D$ preserve vector space operations, i.e. sum and scalar multiplication – Federica Maggioni Apr 22 '13 at 10:38
for $DM-MD=I$ use the Leibniz rule for the derivative of a product: $D(x\cdot f)=D(x)\cdot f+x\cdot D(f)$ and the fact $D(x)=1$ – Federica Maggioni Apr 22 '13 at 10:41
up vote 1 down vote accepted




$$=aM(P_1(x))+ bM(P_1(x))$$


$$=P(x)-xP^{'}(x)-x(P^{'}(x))=P(x)+xP^{'}(x)-xP^{'}(x)$$ so $\;DM_MD=I\;$

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