X1 + 2x2 + 3x3 = 6 2х1 - 3x2 + 2x3 = 14 2х3 - 3x1 +x2-x3 = -2

1. (a) Use definition of linear transformation to determine whether the transformation T:R? → R³ defined as follows is linear. -y T r+ 2y 2.r – 4y (b) Let T : R" → R" be a linear transformation. i. Is it always true that T(0) = 0 where the zero vectors live in the appropriate spaces? Justify. ii. Does S(0) = 0 mean that S is linear? If not, can you give a counter-example? (c) Consider the following matrix transformation T : R² → R² defined as a b T d i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result across the new r-axis, find a, b, c, d. ii. What if T reflects the 2D space across the r-axis first and then rotates 150° counter-clockwise? Do you expect to get the same a, b, c, d? Why or why not?

Q1) Solve the following linear system of equations using Gauss elimination and using Cramer's rule. (2 + 3i)x + (2 – i)y = (1 +2i) (4 + 2i)x + (2 + i)y = 2 а) %3| b) (2+ )х + (1 — i)у %3D 2 4ix + (2 + 4i)y = i Q2 Find all solution for cos(z) = 10

1) Utilize the fourth order Taylor polynomial for e7x at x=0 in order to approximate the value of e1.2 2) Utilize the Taylor series remainder estimate in order to bound the error in the approximation in #1, where the absolute value of the difference between the actual value and the approximation that's been made: