MATH 218 Final with Solution, Spring 2017

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department MATH 218 - QUIZ II Fall 2008-2009 Directions: e This is a partial assessment of what you are learning in this course. Read carefully and write down what you know. ¢ The exam is meant to detect any gaps in your knowledge so that they are corrected. Show your thought process in the minimum number of sentences. « You have 70 minutes. Finish as much as possible. Work quickly but accurately. e Do not forget to write down your name and circle your section number. Section: 5 (@ 9:30) 4 (@ 12:30) Question | Number of parts and subparts | Grade I 2 124 m | 3 /25 It 1 /12 IV 3 /24 Vv 3 /15 Total Grade = /100 I- (a) (12 points) Determine whether the set of all 9 x 9 matrices , Mgg, is a vector space under the following addition and scalar multiplication: A+B=A+B+], al b| — |(k+1)a (k+2)6 c d (k+3)c (k+4)d Note: In the above definition of addition, the plus sign on the left is the new addition operation, and the one on the right is the standard addition operation. and (b) (12 points) Let W be the set of all polynomials of degree exactly 4. Determine whether W is a subspace of the set of polynomials of degree less than or equal to 4, Pa. We 3 pCa) € R | p x) fas degrer 4} = } a+axt ax taxi+a, x such thot ALAL4A, a, aeR and a #0 5 vd Ip xe x’ EW 2-7 _ x EW II- (a) (i) (10 points) Find a basis for the space spanned by the vectors v, = (1,3, 5,9, 7), v2 = (2,4,6, 12, —2), v3 = (3,7, 11, 21,5), v4 = (2,8, 5, 1,0) which is a subset of {v1, v2, U3, U4}. , 2 3 27 5f 1 2 3 27s 2 3 2 34 4 F © -2 -22 eo t | -t 5 6 ;. 5 Oo -y -4 -5 o 4 4 Ss g9 22 5 Oo wv eZ -!) 6 I+ O -16 ~i¢ —I4 o 16 16 [ 62 32 MD 2 2 2 o l 1 —\ fo) @ 1 of ©o Oo Y o eo eo ® © oO oOo 23 0 oe CO ° Oo 0 ?#e o © ° $f 4 ot bw, we 43 4& pu YruUjpua basis fr Spas}, %, 5%} tl 2s (ii) (6 points) Write any vector that is not in the basis as a linear combination of vectors in the basis. Us “tu, So Uy = wtVe (b) (10 points) Find a basis for the space W consisting of vectors (a,b,c) such that 221 +1622 = 0 (a, 6) is a solution of the system { 32, +2422 = 0 —4z, -322,= 0 W = } (a,b,c) har a=-St bot andt A} (abe) (-34,t,<) =t(-8,1,0)+ < Coa) Se 3B), (0,2,:)} Spans W so Nome ofthe tue veehs_, ve ~ midtipl ef 1 the ethan So § (54,0), (oo)){ - baw RW HI- (12 points) Prove that if the rank of an m x n matrix A is m then ColumnSpace, = R™. rank (A) = dim ( Rew Spacy ) = din (Column Space, ) =m Rub CobunnSpote % a suloe pace of R™ and “+ Rao the same dimwien a. hKR™. Therefore, Column Spaeey= IR™ (b, ~ therm) ‘V- Answer each of the following questions. Give justifications for your answers. (a) (8 points) Consider R? with the inner product < u,v >= 3u,v) + 5u;v). Use the Gram Schmidt process to transform the given basis u, = (2,—1), uz = (1,1) into an orthogonal basis. we (2,-1) Us W- He} wy where We Spanhv WwW om uw <4u,,V> Ww ' \ (1,1) — 3x2xl+5eCadal (3 -1) 3x 2m 4 SxG)al) “ = (1,1!) (2,-1) = (2, 1) + 7? 79 \F {b) (8 points) Given a space V with the usual Euclidean inner product and a sub- space W of V with the basis {(1, 2, 3), (3,0, —1)}. Find the projection of (2, —7, 3) on W. me (3, 2,-1)} b regent 4 -43) - “geite| [2,3 o-3 -| 7 f “T+ 444. 4+, @ )+ 23 (30 ) = <2 (12,3) + 2 (3,¢-1) (c) (8 points) Find the orthogonal complement of the space spanned by the vectors (1, 2, 0, 0, 3), (0,0, 0, 1, 1), (0, 0, 3, 0, 1), (0,0, 0, 0, 2) fr Al 2 0 28 o |! \ 3° | o 9 Sv 2 rey oO ) Goo (Row Space y= NM spaer, A oO Lo2 oe o 3&3 jo]—sx|\ 2 , © 7 ° oo 20 | °o o a 4 \ \ © x ze 5 gS mye 5g Nbipag = fxeR ” = © X= (-2t, 4,47 ) et oboe + € m2 72% V- Give a short answer to each of the following. JUSTIFY. 1. ( points for each answer) The nullity of ann x n matrix whose entries are all ones is: | —> ,\--- J oo -— VF _ oao- — Go ™ : co) | ~{ °° _ Le ane Wel pre vevveble, daa (Wall space) = relly of he metros =n-l 2. (5 points for each answer) Can an overdetermined system Az = b be consistent for every possible b? a weckous can not Span RR” Se er. aglim cant be umsisterd Wb. 3. (5 points for each answer) If the vectors u, v, w are three vectors orthogonal toa space W, can you conclude that the space spanned by u, v, and w is orthogonal to W? \ 3 the Cew’> = OoO+0%790 GOOD LUCK 2 Oo Gr ok w’ ce W