For Problems 9-13, show that the given mapping is a nonlinear transformation. 9. T : P2(R) - R defined by T(a + bx + cx2 ) = a+b+ c+1. 10. T : M2(R) - M2(R) defined by T(A) = A2. 11. T : Co[a, b] - Co[a, b] defined by T(f (x)) = x.: " - Rm (en)]. 12. T : R2 - R2 defined by DS n, the formula T(x1, X2) = (x1+ x2, 2). 13. T : M2(R) - R defined by ust satisfy T(A) = det(A). (v) For Problems 14-18, determine the matrix of the given trans- rs c and d. formation T : R" - Rm. tion 6.1.3 that 14. T (x1, X2) = (3x1 - 2x2, x1 + 5x2). 15. T (x1, X2) = (x1+ 3x2, 2x1 - 7x2, x1). 1 - x2). 16. T (x1, X2, X3) = (x1 - x2+x3, x3 - x1). 17. T(x1, X2, X3) = x1+5x2- 3x3. x2) . 18. T(x1, X2, X3) = (X3 - X1, -x1, 3x1 + 2x3, 0). For Problems 19-23, determine the linear transformation T : R" - R" that has the given matrix. (2.0.0.1 n I. 19. A = - 4 20. A = -1 W N 1 NU N 21. A = OO N W UI AN 7 -3 22. A = -ONI 23. A = [1 -4 -6 0 2]. T bos (je) T.bart tr( A), where 24. Let V be a real inner product space, and let u be a fixed (nonzero) vector in V. Define T : V - R by T (v) = (u, v). Use properties of the inner product to show that T is napping is a a linear transformation. 25. Let V be a real inner product space, and let uj and u2 be fixed (nonzero) vectors in V. Define T : V - R2 by T ( V) = ((41, v), (12, v)). = A2. Use properties of the inner product to show that T is f ( X ) ) = X. a linear transformation