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Define the linear transformation S : Pn ? Pn and T : Pn ? Pn by...

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Advance Math

Define the linear transformation S : Pn ?Pn and T : Pn ? Pn by S(p(x)) =p(x + 1), T(p(x)) = p'(x)

(a) Find the matrix associated with S and T with respect to thestandard basis {1, x, x2} for P2 .

(b) Find the matrix associated with S ? T(p(x)) for n = 2 andfor the standard basis {1, x, x2}. Is the lineartransformation S ? T invertible?

(c) Is S a one-to-one transformation? Is it onto? What is thekernel and range of S? What is the rank and nullity of S? Verifythe Rank Theorem.

(d) Is T a one-to-one transformation? Is it onto? What is thekernel and range of T? What is the rank and nullity of T? Verifythe Rank Theorem.

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