MATH 218 Final, Fall 2006

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Final Exam Fall 2006-2007 Name:........................................ ID:............................................. Section: 4 (@ 12:30) 5 (@ 9:30) Time: 120 min Directions: • Read each question carefully before you answer. • If you are asked to answer by TRUE or FALSE, make sure that you do not give an abrupt answer. I- Give a precise definition of the following: (a) (4 points) A vector space: (b) (3 points) A basis for a vector space: 1 (c) (3 points) The rank of a matrix: (d) (3 points) A linear transformation between two vector spaces: (e) (3 points) The nullity of a linear transformation: II- Let A =   1 4 −3 0 2 10 0 4 3 13 −7 3   (a) (2.5 points)Find a basis for the row space of A. 2 (continue your answer here) (b) (2.5 points) Find a basis for the column space of A. 3 (c) (3 points) Use (b) to show that the system   1 4 −3 2 10 0 3 13 −7   x =   0 4 3   is consistent, without solving it. III- Let B =   1 1 1 0 2 2 0 0 3   (a) (4 points) Find the eigenvalues of B. (c) (8 points) Find bases for the eigenspaces of B. 4 (Continue your answer here) (b) (5 points) Is B diagonalizable? If yes, find a matrix P that diagonalizes B and determine P −1BP. IV- Consider the system Ax = b. (a) (i) (6 points) Find a least squares solution for the system if: A =   2 1 −2 1 0 −1 1 1 0 1 1 −1   and b =   6 3 9 6   5 (Continue your answer here) (ii) (4 points) Find the orthogonal projection of u = (6, 3, 9, 6) on the subspace of R4 spanned by the vectors v1 = (2, 1, 1, 1), v2 = (1, 0, 1, 1), and v3 = (−2, −1, 0, −1). (b) (3 points) If b is in the null space of AT. What is Proj W b ? Justify. 6 V- Let T be a function from the vector space of all n × n matrices Mnn to itself with standard addition and scalar multiplication. Let C be a fixed n × n invertible matrix. Define T by T(A) = CA for all A ∈ Mnn. (a) (4 points) Show that T is a linear transformation. (b) (5 points) Show that T is one-to-one. (c) (5 points) Deduce rank(T). 7 (Continue your answer here) (c) (3 points) Is T is an isomorphism? Explain. VI- Let T be a linear transformation from the vector space P2 to the vector space R2 with standard addition and scalar multiplication defined by: T(p(x)) = ( p(0) , p(−1) + p(1) ) (a) (5 points) Let B = {1, x2, x} and B′ = {e1, e2}. Find B′[T]B. 8 (b) (5 points) Define, then find ker(T). VII- (8 points) If T : V −→ W is a one-to-one linear transformation and {v1, v2, ..., vn} is a basis for V , then {T(v1), T(v2), ..., T(vn)} is linearly independent in W. 9 VIII- (5 points) Specify the appropriate vector space, operations and inner product , then, use Cauchy Schwarz inequality to show that for all positive numbers a1, a2, ..., an, (a1 + 2a2 + ... + nan) � 1 a1 + 2 a2 + ... + n an � ⩾ n2(n + 1)2 4 10 IX- (6 points) Write TRUE or FALSE. 1- If λ is an eigenvalue of an invertible matrix A, then 1 λ is an eigenvalue of A−1. 2- If v is an eigenvector of a square matrix A, then it is also an eigenvector of A5. 3- If det(A) = √ 3 then λ = 0 is an eigenvalue of A. 4- If T : R5 −→ M33 and rank(T) = 5 then T is not one-to-one. 5- The matrix ATA − 3I is orthogonally diagonalizable for any matrix A. 6- If {v1, v2, ..., vn} is an orthogonal set of nonzero vectors in an inner product space V such that dim(V ) = m , then m ⩽ n. GOOD LUCK If you have any comments or requests, please write them here: 11