Problem:
Linear Mappings, Differentiation and Linear Spaces
Please help with the following problems. Provide step by step calculations for each.
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 counter example:
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)!