MATH 218 Final, Spring 2012

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 218 - Final Exam Fall 2006-2007 Section: 4 (@ 12:30) 5 (@ 9:30) Time: 120 min Directions: e Read each question carefully before you answer. e 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: No defini tons in this semester’s Grad but make SuYve that gow know thern, they Aare, ° strenrely im portant . Vou can’t solve problems i§ you do not know what they ave talking about . (b) (3 points) A basis for a vector space: (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 14 -3 0 A= |2 10 0 4 3 13 -7 3 (a) (2.5 points)Find a basis for the row space of A. —> (continue your answer here) | 4 -3 im ° 2 lo oO ? 4 3°13 ~-4F 2 3 35! -2 6 "ge 1 3 5 ’ 7 8 o CD : 6 ' 2 3 oe O°! Therefore, (b) (2.5 points) Find a basis for the column space of A. From tee REF of A above , & basis Column Space of A is a. |=): ? 13 (@ineon vectors of A conrearapen ditva by Catn yr Wee dors ao , +he K E Foe cent ue, a fea ive ne ! > toy lt the (c) (3 points) Use (b) to show that the system 1 4 -3 col ~|]2 10 O|x =1/4 cual 3 13 -7 is consistent, without solving it. FT} Ppa es ee & colle Lea be pee ~ atux A. Sey ie) as} Spee * contd oy j [B}> & Spee ADD) Da ]he ree Thosfore cue 4 BAL B X= L > b € le bamnSpoees IH- Let (a) (4 points) Find the eigenvalues of B. oi Is Ca egenvalrre £6 c=) det (XL - B)= Oo <> |’s -| =} -1 c=? fray Q-2) (2-3) = ° = Oa Q ee nd (c) (8 points) Find bases for the eigenspaces of B. Eigen Space =} xe rR? lAy = dv { = MAD S pace a, VIA 0 -. -llo"}> fe i tye wet 0 -| -9 oO oO Qo { for] X,+ Oo 0 QO ~2} 2 gd 9 9O;\9 =e Care S Pre R? | X= (42,0) whe te RI Thess, (10,00) & a bart Eger Spm (Continue your answer here) EC igor Spree » fv dd prer, L- A T -} -| 2 + p -!-iJoy—> |! {9 aa 4 Oo 0 -2/0 | o 9 5 “4 oo -1}0 po 2 8 8) “* bases an , (11,293 S \ D Comte Spa - Nabe id 1A Fae Pp Lae. a, | g _t a \ t ¢ F 2 4 1) ‘ aw a 7 \ se 2% é D (b) (5 points) Is B diagonalizable? If yes, find a matrix P that diagonalizes B and determine P=! BP. 3 de 4 ned 7 ta ew Pathe, So € A Ab cues oral’? ably ¢ ! a Poy L ol 4% | 0 \ > | é CO | | -! { o oO PBPe| 34 3 o 8 2 IV- Consider the system Az = b. (a) (i) (6 points) Find a least squares solution for the system if: 21 -2 6 10-1 3 A=) 1 g| andb= |g 11-1 6