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

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

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

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

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

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

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Expert Solution

Colby Messinger answered 1 year ago

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