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Linear Mappings, Differentiation and Linear Spaces

1) Show that this mapping is linear:
T: P5 -> P8 defined as Tp(t)=p(t+1)-p(t)+integral(t-1 to t) s^2 p(s) ds

2) Prove the following is true, or give a counterexample:
If l is a nonzero scalar linear function on linear space X (which may be finite or infinite) and a is an arbitrary scalar, there exists a vector x in X st l(x)=a

3) Let T: Pn->Pn be the linear map st Tp(t)=p(t+1). Show that if D is differentiation then T = 1 + D/1! + D^2/2! ... + D^(n-1)/(n-1)!

Solution Preview

that T maps P_5 into P_8 should be clear on account of the integral;
and for all scalars c, and 5th degree polynomials p,q

T(cp +p') = (cp + q)(t + 1) - (cp + q)(t) + int_{t-1}^t s^2 (cp + q)

= cp(t + 1) + q(t + 1) - cp(t) - q(t) + c int_{t-1}^t s^2 p(s) ds + ...

Solution Summary

Linear Mappings, Differentiation and Linear Spaces are investigated.