For each of the following linear transformations T, determine whether Tis invertible and justify your answer. (a) T: R2 R3 defined by T(a1, a2) = (a12a2, a2, 3a1 + 4a2). (c) T: R3 R3 defined by T(a1, a2, a3) = (3a12a3, a2, 3a1 + 4a2). (d) T: P3(R) P2(R) defined by T(p(x)) = p'(x).

Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be su you can justify your answers. A. T(A) det(A) from R³x³ to R B. T(A) 1A from R4x3 to R4x3 OC. T(A) = AT from R5x3 to R³x5 1 D. T(A) = A 1 from IR to R2x2 6 8 JE. T(A)= A2 from R4x4 to R4x4 6 5 R.T(A)=[74 -97 A from R2x5 to R2x5

Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers. A. T(A) = trace(A) from R³x3 to R OB. T(A) AT from R4x5 to R5x4 0 2 C. T(A) = A [²3] A from R2x2 to R²x2 8 6 OD. T(A) A7 from R6x6 E. T(A) = SAS-1 from R2x2 to R2x2, where S = OF. T(A) det (A) from R5x5 to R to R6x6 -7

X1 – 2x2 -3x2 [2x1 – 5x2] Determine if the transformation T is linear or not. If it is, prove it. If it is not, find a counterexample.

Determine whether the following are linear transformationsfrom P2 to P3. L (p(x)) = x2 + p(x)