MATH 218 Final, Spring 2010

American University Of Beirut Mathematics Dept. Math 218 Final Examination, Spring 2010 Time: 2 hours Instructor: Michella Bou Eid Section: 1 __________________________________________________________________ Name:________________________ ID:____________ Maximum score: 100 pts (+3 Bonus pts) Exercise 1: ( 18 pts) Let (i) (7 pts) Find a basis for null(A) and deduce its dimension. (ii) (6 pts) Write a basis for each of the column space of A and the row space of A. Do not Justify (iii) (2 pts) Check the rank nullity theorem for A. (iv) (3 pts) What is the dimension of Exercise 2: ( 18 pts) let (i) ( 6 pts) Find the eigenvalues of A (ii) (7pts) Find a basis for each eigenspace of A. (iii) ( 3 pts) Find a matrix P diagonalizing A and the corresponding diagonal matrix D. What is the relation between A, P and D. (Do not compute or check your answer!). (iv) (2 pts) Write the formula for calculating Exercise 3: (10 pts) Consider the inconsistent system Ax=b where i) (7 pts) Find the least squares solutions of the above system (ii) (3 pts) Deduce from part (a) the projection of b on col(A) Exercise 4: (12 pts) A. Let W = span , be a subspace of and let v = be a vector in i. Find an orthogonal basis for W using Gram – Schmidt process ii. Find proj w v B. Prove that finding the least squares solution by solving AX = is equivalent to solving AtAX = Atb Exercise 5: (10 pts) Show that the following linear transformations are isomorphisms (i) (ii) (Hint: Write ) Exercise 6: (12 pts) (i) Let T: V V be a linear transformation. If {T(v1),…..T(vn)} are linearly independent, show that { v1, ….,vn} are linearly independent in V. (ii) Prove that similar matrices have the same determinant & the same eigenvalues. (iii) Show that if A and B are nxn diagonalizable matrices with the same diagonalizing matrix, then AB=BA Exercise 7: (20 pts) Prove (briefly) or disprove ( explain or give a counter example) a) There exists a 1-1 linear transformation T: V W be a linear transformation if dimV=7 and dim W=5. b) The polynomials are orthogonal vectors in the inner product space where c) In any inner product space d) If A is diagonalizable, then are also diagonalizable e) If Bonus (3 pts): State without proof 3 properties of real nxn symmetric matrices