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Linear space

Suppose S is a linear space defined below. Are the following mappings L linear transformations from S into itself? If answer is yes, find the matrix representations of formations (in standard basis):

(a) S=P4, L(p(x))=p(0)+x*p(1)+x^2*p(2)+X^3*p(4)
(b) S=P4, L(p(x))=x^3+x*p'(x)+p(0)
(c) S is a subspace of C[0,1] formed by Span(xsinx, xcosx,sinx,cosx)and L is the differential operator: L(f(x)=f'x(x). Note use the vectors forming the span as basic vectors

Solution Summary

This shows how to determine if mappings are linear transformations and find the matrices of the transformations.