Question: Problem 6. Consider the following functions between vectorsoaces (with their usual operations) over it . Decide if the function is a linear transformation or not.

Problem 6. Consider the following functions between vectorsoaces (with their usual operations) over it . Decide if the function is a linear transformation or not. if it is linear, prove it, if not; demonstrate why not. (4) T : P2(R) - P2(R) , where T(p(x)) = p(0)x2 + 1 (5) T : M2x2(R) - R , where T(A) = det (A) (6) T : M2x2(R) - R , where T(A) = tr(A) , where tr a b =atd , the sum of the diagonal entries. C d (7) T : M2x2(R) - M2x2(R) , where T(A) = QAQ ! , where Q = 4 (8) : 0 = 0% , where T(f) = f' -2f . Here " is the set of all functions that are infinitely differentiable on the real line.Problem 7. Suppose T: P(R) - R' is a linear transformation, where T(1) = " I ( 1 + * ) = , and T(1 +x+ (1 ) What is T(0) ? (2) What is T(atx?) ?Problem 8. m+2y (1) Suppose 1':t2+1R3 is linear where 1'[:] == 3y , find the standard matrix 21 for I\" , where Titan

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