MATH 218 Exam 2, Spring 2018

April 3, 2018 MATHEMATICS 218 NAME Kew Ly on. | ) Time : 55 minutes QUIZ II [DH wwneeene eee en nee ee Spring 2017-18 Circle your section number : Michella Bou-Eid Sabine El Khoury Giussepe Della Sala Hazar Abu-Khuzam Rana Nassif 1 2 3 4 5 6 7 8 5 10 11 12 13 14 15 8M 11M 2M 12M]3M | 4M] 8F] 11M | 11F 2T 3:30T | 5T}| 1M | 3M | 4M PROBLEM GRADE PARTI ee /16 2 0 /16 a /16 foe 18 PART II 516/17/81}9 | 10] 11 | 12 alalalala {ala la (DOK! b |b fb [b |b clcelelc ele [ec is) d{d[d kad |d Mla elelelele fle le 512 ------- / 32 PART UI 13 14 15 16 17 18 or te Pert > 13-18 ------- / 12 TOTAL — — ——wwre=== / 100 PART I. Answer each of the following problems in the space provided for each problem ( Problem 1 to Problem 4). 1 0 3 2 1 6 1. LetA= 1 0 3 20 6 (a) Find a basis for the null space N(A) of A. (Do not prove it is a basis) Pyer [8 points] i . Z10 points 1 a Sy 2K tke o oO\S a \ eg {eo —> |\o (Aj {( © 342 Kae so Goa \o 2 @ C10 =2R 2 > & 0 SO : \ fe ‘ vk d mo Veto tteAwR (b) Find a basis of the column space Col(A) of A. (Do not prove it is a basis) [8 points] Columns LK 2 in rov-edolon foun coker Pole => Correapen : Cobimund \ % Q wm Po Kye oivurx Loum ce if arts of The Cybern Y a Cok (A) 2 toe | LW = 4 (2),(8) 2. Prove or disprove the following statements: (a) Let W={p(x)eP, | xp'(x)+1= p(x)} . Then W is a subspace of P, FALSE The Zt phnaml BR, PW)=0 ¢W AfnG_ LPC) + lax O41 = | 4@ ( Com allen ive an exangrle Show yee) [8 points] a (b) ee b| er] b=(atc)/2 }. Then V is asubspace of R°. qeLE Fy é . dine (e eV [8 points] 2. CRosed ander addin Non | pie alec! f (3) XG) eV, De bee (al “ {& oil s t : £\. () vba barbs se + oT Se (i) (% ced © (ore) a NV ; 3. Consider the subspace W ={p(x)eP, | p(l) =0} of P,. Let B={x-1, x’ -1, x°-1} (a) Prove that B is linearly independent _ cS [8 points] | cy (x1) + Co 0-1) + (1) _ | | : cs) =O ceded 42 * 7) = aX as Cx ey x Cc, —C,- ‘al a — C2 CC, = Seo BB So 6 (b) Prove thatBisabasisforW _ B CW sin Pleo Yee Enough Ts Shaw I hod Spee ae Pere art bee ox B , +d eW , Wee a cW 7} Ow B AN Fors 2 Pi Puiys arbreta = 0 This sfours Lik” These vecbos wats wee 8, Beet Gee Gea tes, wae WN 8, Os yet “De Spel®) eno B02 3 \\ablee') 4 o& : gad PH)=% (x Se Spor (3) = \ am yeu =) B = 3 ‘ wh oes hens hn \W/ 214 ta) ) 1 1 linear transformation. (That is show T(v)=0 for every v € R’) +(s)eT(-Mar )=TU)=2 ev TE)=T(C)-Q)= TE) TM) 1 2 1 4. If T: R? ——> R isa linear transformation with 7 = if -1(;) . Prove that T is the zero PART I. Circle the correct answer for each of the following problems (Problem 5 to Problem 12) IN THE TABLE OF THE FRONT PAGE . [4 points for each correct answer (No penalty)]. 5. Let A be mxn matrix and let R be a row echelon form of A. Which one of the following is FALSE? (a) N(A) = N(R) ((b)} col(A) = col(R) (c) Ifcolumns 1 and 2 of R are linearly dependent then columns | and 2 of A are linearly dependent. (d) The nonzero rows of R form a basis of row (A) [ 4 points] 1 -2 6. The subspace W of R* spanned by S= {| | |, 3 \ S ( \ y 4 g, © 4 (a) 1 2344 rete (533 € C&) 2 3 Vu S fap & ao \ \ () 3 ee ee ONT Ooo (d) 4 (e) None of the above — JT [4 points] 7. Let T:R?-R’ be a linear transformation such that aI ; } 1 land { / 1 1 nm of 2} 1(G)-0)) \\ af! ; = T(\)- rs o(3} = ay-Q) 3) -G) -1 ©) | 2 (d) | 2 1 (e) None of the above 4 points 8. Let S={x+1, x°+x, x°+x?}c P3. Which one of the following is TRUE: (a) Sis a basis of P3 (b) Sis linearly dependent (¢) the constant polynomial 1 belongs to Span(S) . Orwearbe snl dt (d)) Span(S) is a subspace of P3 of dimension 3 3 7) e ) None of the above [ 4 points] 9, Let S={u, v, w, z} be a linearly independent subset of a vector space V. Which one of the following statements is FALSE: (a) The dimension of V is at least 4 (b) The set{u, v, z}is linearly independent x) Span(S) has dimension 3 S is a basis of Span(S) [4 points] 10. Let A be mxn matrix with m<n. Which one of the following statements is FALSE? (a) The columns of A are linearly dependent. . (b) col(A) is a subspace of R™. {2 3 23 (c) row(A) is a subspace of R". ( 24 4 é d) dim(N(A)) # 0 cB The rows of A are linearly independent. [ 4 points] 11. Let W ={p(x)eP, | p(1)x’ + p(-l)x + p(0) = 0}, then dim W = (a) 3 Pex) = aX bx+e A ph why eC. ar PCijz ate bee » PC A)=a-b4 L Cc wt, f +-C ~~ = O @® 0 (arbre) x (a-bt One = S (e) None of the above at barca” = weak = _— a—-b tt =2 c= 0 [ 4 points] 12. Let U be the subset of P3 given by U={p(x) = ax? +bx? +ex +d €P, |p"(x)=0}. Then dim U= POX) ~ 3a xr4 2boKxt eo ’ Px) a2 Gax +2b =0 oO => aso, bao 1) 3 (e) 4 [ 4 points] PART UI. Answer TRUE or FALSE only: (Questions 13 to 18) IN THE TABLE OF THE FRONT PAGE [2 points for each correct answer, NO PENALTY] { 13. If A and B are two nxn matrices such that AB = 31, then Col(A) = R". - 14. If S= {v}, then S is linearly independent. of vane, Then Se y ° @ eW E 15. A subspace W ofa vector space V is linearly independent. — 2 T 16. The set of polynomials {1+x, x°+3, x?+x} is a basis for P,, Cita) 6 ie “TY” 17. IfA is a 3x5 matrix, then the columns of A are linearly dependent 5ved a mm | ee ‘ —_— 6 ; a E 18. T: P, ——>R defined by T(ax”+bx+c)= lal isa linear transformation | asl at [12 points] “7 C 4 } i uw im rode 4 qT cx) glee