MATH 218 Exam 2, Fall 2016

October 27, 2016 MATHEMATICS 218 NAME Time : 65 minutes QUIZ 2 ID# Fall 2016-17 Circle your section number : Giuseppe Della Sala Sabine El Khoury Monique Azar Monique Azar Hazar Abu-Khuzam 1 2 3: 4 5 6 Ui 8 9 10 i 12 13 14 15 2W 1W 11W 12F | 4F oie 12M ]1M | 11M | 4M 3M | 11Th 11F | 4F SF PROBLEM GRADE = 1 |6 /16 eee) insight ey bal Se insight et TOTAL 16. /100 | PARTI. Answer each of the following problems in the space provided fo problem ( Problem 1 to Problem 4). 1. For each one of the following vector spaces V and subsets WCV, prove or di W is a subspace of V. (a) V=Mow, W= {AE Mae | AA'=A}; ro > (M8 & Pe SS. i: Fo foB-0 y! a. (0 y J iG nie) © 2ere palpated On eon Laie OeEwWs Ww ciel 2 Let PEW EL FaEW =D>( xp tx)-tple) O\ 7 NT rata) -2gli)=0 |SUrfast L(preg) (4) -2(pr Gta a +) pt) + (4) » | -2 J ec) rene) | = tfee)reqts) J 2 pt) +9) ins are 2 pita) renal) 0g) RD | ig = J upla)-2ptx) ) + ¢ fray) - 29x) } 2 = [o} #c)o =O => oe addition 4 scablat” mudfigica HO = Soy W is subsfae af fe. 2. Let A be the matrix 1 A= 2 1 (a) Find a basis for the null space N(A) of A. (no need to prove it is a basis) - Takelrx- a be ig He an of AX 20=> cugmented ee % ME {eet ° | (e} (eo) o It -l “RtRR3 fo | = Hb : i é | _— tet is afiwe ie ves So, Ha-t=O = Aart a % ee ap melt Soy MA)= es ) per }- fe (3 i} [ech ob ae -f NCA) (b) Find a basis of the column space Col(A) of A. (no need to prove it is a basis) wie wee te —" cele bs [8 points] wif fay NwWe Orn © kL u | insight insight | => meats "7 ‘) which anc the = hrs Puan of A (initially) . insight er lL. U 3. Show that Span{1, x, x}= Span{ x?-x, x+1, 5} in Py oS >t 6 points} ) - Span flat} = at tala tale) |INSIQhk ec LU b HQ ter Gee %) - Dis Zf fs =) {x 0, Span a gwice = tlgX 103% < a+bx oat 1 ° ° Spon fra, UlS { = C2+5eg + (-cltch) x a z, ‘ => 22H, 2t15 spon R since chr5ck (el tch) tela Se, Span(lxx"} =Span fe xel, st= ¥,. Gs 4. Let V be a vector space such that {uy, u2, u3, u4} is a basis of V. Let a b=-</ teh U=Span{uy, u2 }and W= Span {us, us}. Show that UNW={0}. oF Cac! ‘a \ 3 i . Put ear sty ¥ akon AV [ 8 points] =P WALD KY are Liveathy independont- => Guy, Ho2uztCzXz tlyky =O far orky for yoy Pt gaty a LE a a Fa ORIG OE ET ea a + WoSpan Sus uy} —— aaa Eig +b Uy > u' where u'e Wl £4,ER + Lug +Louy = Oo Z Se ONW =O PART IL. Circle the correct answer for each of the following problems (Problem 5 to Problem 10) IN THE TABLE OF THE FRONT PAGE. [5 points for each correct answer]. 5. Let U be the subset of P, given by U={p(x) = ax’ + bx? +x? +de +e EP, | p(0) = Oand p"(x) = 0}. insight 6. Let S={u, v, w} be a linearly dependent subset of a vector space V. Then (a) w is a linear combination of u and v (b) {utw, u+v+w, v}is linearly independent in V (c) u=0 or v=0 or w=0 © {2u, v, 3w} is linearly dependent in V. (e) None of the above “sight 7. Let T: RR bea linear transformation such that 7(3)=5 , then T(8)= co (a) 8/3 b) 5/3 40/3 (d) 3 (e) none of the above. [5 ppi a E 8. Let Uand W be the subspaces of r° defined by: ins | L S l e a 0 Cc U={{b| ER] a=b=c} , W={| y| ER) y,zeR}, then dim UNW= € Z) a.l b,2 c.3 @o0 e. none of the above. [5 points] 9. Let V be a vector space of dimension n. Which one of the following statements is FALSE: Any set of (n+1) vectors in V is linearly dependent. T; a. b. Any linearly independent set of n vectors in V is a basis of V. @ If U and W are subspaces of V such that dim U=dim W, then U=W . Any set of n vectors spanning V is a basis of V. @ If U and W are subspaces of V, then UNW is linearly dependent 10. Let S={v1,v2,v3}C R‘ be a linearly independent set. Which one of the following statements is insight Cy pomesy (a) The set {v1,v3} is linearly independent. (b) The subspace span(S)CR* has dimension 3. span(S)=R* @ The set {2v1,V2,v3} is linearly independent. (ce) For any vector v€span(S), the set {v1,V2,V3,V}is a basis for Re, [5 points] PART III. Answer TRUE or FALSE only IN THE TABLE OF THE FRONT PAGE [3 points for each correct answer, -1 point penalty for each wrong answer] 11. If {v1, v2, v3} is a linearly independent subset of R®, then {vi+v2, VirtVatvs, V3} is linearly independent. 12. 13. 14. 15. 16. 17. 18. The set of all 2x2 non-invertible matrices is a subspace of M,,,. £ x Let V=J| y ER| x,yER\. Then Vis a subspace of R2.Y | The set of polynomials {1+x, x°4+3x+3, x’} is a spanning set for PA If V is a vector space having only finitely many elements, Then v={0}.7 a Let W= (6 b at+2b JeMan | avert then W is a subspace of M,,, T Let A be an mxn matrix. Then the set W={rER"|Ax = of is a subspace of R". “| If T:V—W is a linear transformation and if {uy, uz, us} is a linearly dependent subset of V, then {T(u1), T(uz), T(us)} is linearly dependent in W. oo = insight [ 24 points} —~ c= INSIGNT] —