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# Linear space

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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.

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
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• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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