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MATH 221 - 102 EXAM #2 Name: Student ID: Exam rules: • No calculators, open books or notes are allowed. • There are 5 problems in this exam. All problems are worth the same number of marks. • Some problem contain several parts. Please read the problems carefully. • Use opposite empty pages if needed. 1 Problem 1. Let R : R2 → R2 be the rotation by 60◦ angle counterclockwise, and let S be the transformation ¸ · ¸ · 2x1 − 3x2 x1 . = S −x1 + 2x2 x2 Find the standard matrix of the composition T : T (~x) = R(S(~x)). Problem 2. Let 0 1 2 −3 7 1 1 3 −3 6 A= 1 −1 −1 3 −7 . 2 0 2 0 2 (a)Find a basis for the nullspace of A and a basis for the column space of A. (b)Determine if the linear transformation defined by A is one-to-one or onto? Problem 3. Let 1 −2 2 A = −2 5 −6 1 −4 5 (a)Find the inverse of A. (b)Find the inverse of AT · A. Problem 4. Find the determinant of the matrix ( 12 A)−1 , where 1 2 −3 4 −4 2 1 3 A= 3 0 0 −3 . 2 0 −2 3 Problem 5. Answer each question ”True” or ”False”. No explanation is necessary. (a)If AB + BA = 0, then A2 B 3 = B 3 A2 , where A, B are n × n matrices. (b)If A is a 5 × 2 matrix then the transformation S(~x) = AT A~x is linear. (c)If a linear transformation T has standard matrix A of size 4 × 5, then T can not be onto. (d)If AP = P B for some invertible matrix P , then A and B have the same determinant. (e)When A, B are n × n matrices, then the set of all vectors ~v such that A~v = B~v is a subspace of Rn . (f)If A is a 6 × 4 matrix such that its nullspace has dimension 2, then the linear transformation defined by A is onto.