MATH 218 Final, Fall 2009

AMERICAN UNIVERSITY OF BEIRUT MATHEMATICS 218 FALL SEMESTER 2009-2010 FINAL EXAMINATION VERSION A Time: 120 minutes Date: January 27, 2010 NAM! ccceeeseceesceesseeecseeeeseecuseeccseeesseeecseeesesesusesesuescsecesrecceseaeeersevertaeeess Circle your section number in the table below: Instructors | Azar } Egeileh | El Khoury | Fuleihan | Karam | Nassif Section 8 5 1 6 4 2 Section 3 9 7 QUESTION GRADE 1 /25 2 [25 3 [25 4 /26 5 /30 6 /34 7 /35 TOTAL GRADE /200 Answer the following seven sets of questions; the back of pages may be used as scratch. 1. Let T : R3 > R# be defined by: 22+ y + 3z t z+ 2y T y = ves (a) Show that T is a linear transformation. (b) Show that T is one-to-one. (c) What is the rank of the matrix representing T. (d) Find a basis for the range of T. (8 points (6 points (4 points (7 points ) ) ) ) 12 -1 #1 2.Let A=] 2 4 -3 3 3.5 -6 —5 (a) Find a basis for the N(A). (10 points) (b) Determine a basis for N(A)+ with respect to the Euclidean inner product. (7 points) (c) What is the dimension of N(A?)? (8 points) 1 -1 -1 3. Let A=] -1 1 -!1 -1 -1 1 (a) Find the eigenvalues of A. (10 points) (b) Find P that diagonalizes A. (10 points) (c) Deduce P=! AP. (5 points) 4. (a) Without solving, find condition(s) on @ and @ for the following system to have exactly one solution: (10 points) at+yt+az=1 r+yt+fz=2 axz+ By+z=-1 (b) For a = @ = 0 find the least squares solution(s) of the linear system. (16 points) 10 5. Let W = {p(2) € Py/ 2p(2) - p(1) = 0) (a) Prove that W is a subspace of P3. (8 points) (b) Find a basis for W. (12 points) (c) Let T : P3 > P, be the linear transformation defined by: T(p(z)) = xp(2) — pd) Show that T is onto. (10 points)