MATH 218 Exam 2, Fall 2017

\fa- November 13, 2017 MATHEMATICS 218 NAME Key aacnene Time : 60 minutes QUIZ 2 1D Pa ee Fall 2017-18 Circle your section number : Hazar Abu-Khuzam Sabine El Khoury Sabine El Khoury Michella Bou Eid 1 2 3) 4 5 6 7 8 9 10 11 12 2W 1W 11W | 2M 1M | 11M | 4M |3M | 2Th 11F | 4F SF PROBLEM GRADE PARTI hel dtc /16 ees /16 . 19 —— 17 PART II 516|7|8|9 |10|11| 12| 13 a ajaja a a a a a violblbib |b |b |b |b cjejo|;jcjc c c Cc Cc dldldld|d|[d [ad |d [a ej;jeyjyetsele € € € € Sk eee / 28 PART III 14 15 16 17 18 19 T T T ie Ti T F F F F F F es mes / 24 TOTAL /100 PART I. Answer each of the following problems in the space provided for each problem ( Problem 1 to Problem 4). 1. LetA= 2 4 6 (a) Find a basis for the null space N(A) of A. (Do not prove it is a basis) : ° : ' | 2 i +2|8 ale [ 8 points] 0 -l -2)e : all mee 2: Y 6leo “RK 3 \ 2 Se bat => ya-2b a-at —>x=-y-e =flt-bt—-t& Nuk spac. 4 Az NCA) | { = ] (Ex) | 4 ty a nO Basis of NA) =} (4)f j Gt)-"(2) \ (b) Find a basis of the column space Col(A) of A. (Do not prove it is a basis) , iS j [8 points] Belanuns (> Y GD wm sew a foe ‘ Entaun Prob So eplaees Ox pain Vhs supe Me we Jom Sats st om 8) \ \ Se Barns | Ot (A) a (2) ) (3 72, uy 2. Prove or disprove the following statements: (a) Let W={Ae Mo, | A is a symmetric non-invertible matrix}. Then, W is a subspace of Mox2 . FRLSE | [8 points] Exch (1 2) ew y e 2) eW (b) Let U={Ae M33 | A is a skew-symmetric (At =-A) matrix}. Then, U is a subspace of Ms,3 of dimension 3. RUE «. Us pp snc Qe [8 points] 3. Let V be a vector space and u, v, w, are vectors in V such that Span{u, ut+-v}= Span{ 2u, 3v, 3ut+2w}. Show that w is a linear combination of u and v Se (Sux 2w) & Spew 5 juss =>) Qo = Ce + C) —~3)hy av > We (c+ co 3) U4 7G erin + u“ a V [ 9 points] Se Wow oo Liyoer Ce 4. Let T:V = V be a linear operator on a vector space V such that T(I(v)) =T() for all veV . Prove that every vectorv <« V can be written as a sum v=u+w, where ue N(T),andwe R(T) (Hint: Find T(T(v)—v) ] WV veV , T(r) =TW) [7 points] => T (TW-v) _ (Tw) —TW) = WW) —Vlv) ag Sy ee we eG So t= VTW)I-V © NT) A aS) oy eee 3 aq veer im N(T) => va Pe care he 4, a vedor a Rly) WT re) PART II. Circle the correct answer for each of the following problems (Problem 5 to Problem 11) INTHE TABLE OF THE FRONT PAGE . [4 points for each correct answer (No penalty)]. i 2% 6°38 5. LetA=|}2 4 0 2 2 2]|.Then 63 3 ue huey |e 2f.2\ (a))Rank (A)=2, Nullity (A)=4 Se o 8 \ ‘i ioe \ Rank (A)=4, Nullity (A)=2 Lig+k: “yg A nc (ey Rank (A)=3, Nullity (A)=3 22> |e 6 -2 ~lo X14 (dy Rank (A)=1, Nullity (A)=5 Ot POOR ak (e) Rank (A)=5, Nullity (A)=1 eae e [ 4 points] oe Rewk (A) ai 6. Let T:P, > R? bethe linear transformation defined by : 3 (G2) T(p(x)) = 2 ie Then dim N(T)= 0 TD) BON igh af vat eee e NL 1 Thos ) = tae (° 2 (d) 3 (e) None of the above des ;, 3-64 tS [ 4 points] 7. Let W={p(x)eP, | p() + p(-l) = 0} .Then Plc )= xp bK+C (a) B={ x?+1} is a basis for W (b) B={ - x?+1} is a basis for W RU) + Pe =e bee 4¢a-b+e B={ - x?+1, x} is a basis for W acral ) B={ x?+1, x} is a basis for W HK Qe4+r2ac = - (e) none of the above [ 4 points] —>cs iss 8. IfT: R? > R isa linear transformation such that al ; ) =l,thn — of x De Vox (a) N(T) = {0}. Aue WOT AE a (Om CO) RT)= R. 6) Dim (N(T))=2. \ (4) N(T)=R?. (e) none of the above. [ 4 points] 9. Let A be a 4x4 matrix and let R be a row-echelon form of A. Which of the following is FALSE? (a) dim (row (A) ) = dim (col (A)) (b) N(A) = NQ@) (c)_ row (A) = row (R) col (A) = col (R) 10. If A is any 3x4 matrix, which of the following statements is FALSE? [4 points] (a) The columns of A are linearly dependent. (b) Row (A) is a subspace of R‘. €)\The rows of the matrix form a basis for row (A). (d) The system AX = 0 has infinitely many solutions. [4 points] 11. Let Ube the subset of P3 given by U={p(x) = ax? +bx* +ex +d €P, | p(0) = 0and p"(x) = 0}. Lae Ple)=o =p d=0o cory Px) = 30K 4+ 2br ac @ 3 P(A) = Laxg2zb so 9 a=e, b=O Ble Qe PY)= oly, oe [4 points] PART II. Answer TRUE or FALSE only (Question 12 to 19) IN THE TABLE OF THE FRONT PAGE [3 points for each correct answer, -1 point penalty for each wrong answer] je 12.If U and W are subspaces of a vector space V with dim U = dim W, then U = W. | 13. The polynomials 2+3x, 3-4x3, x+x?, 1+x-x? , 5x +2x? are linearly dependent in Ps a+b a+b iS 14.Let W= eM, | a,b € Re, then W is a subspace of M,,, of 2a+2b -—a-—b dimension 2. j 15.If T:V—>W is a linear transformation and if {T(us), T(uz), T(us)} is a linearly independent subset of W, then {u1, u2, us} is linearly independent in V. x E 16.Let V=4| y |e R}| x, y € Rp. Then Vis a subspace of R°. 2 x ] 17.The set of polynomials {1+x, x?+3, x?+x, 5} is a spanning set for P2. 18.If A is a 3x3 matrix such that A? = 2] then the rows of the matrix A a 7 form a basis for row (A). A Ca inverbd i 19.If A is a 4x6 matrix with rank A = 3, then dim N(A)= 1 [ 24 points]