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111-2008&2009-3-M20-July2009

111-2008&2009-3-M20-July2009 - Kuwait University...

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Unformatted text preview: Kuwait University Department of Mathematics and Computer science MATH 111, LINEAR ALGEBRA 30 July, 2009, 6—7.30pm. Second Exam. Answer all questions. Calculators and mobile phones are NOT allowed. 1. (2pts) Suppose ”U“ = 2 and “V“ = 3. Is it possible that U—V = —T? (Justify your answer]. 1 2 0 1 0 1 —2 1 2. (3pts) A = 0 0 3 2 —1 0 0 1 (a) Find the cofactor A11. (b) Evaluate |A| using cofactor expansion. 1 (c) Use part (b) to evaluate EodflA) 3. [3ptsj Show that adj(AB] = adj[B)adj(A), where A and B are nonsingular n x 1:. matrices. 4. [3pts] Find the point where the line through the points A(3, —4, —5) and 8(2, —3, 1) crosses the plane 25': + y + z = T". 5. [4ptsj (a) Determine whether the points A(1,1,1),B(2,—1,3) and C(0,3,3) are on the same line. (b) Determine whether the points P(—1, 0, 2}, Q{1, 2, 0), R[—1, 1, 3} and 3(2, 0, —4) are on the same plane. 6. (4pts) (a) Let Q = {($,y,z] G 323: 2.7: = y+ 2}. Is 9 a subspace of 9113‘? (Justify your answer). (b) Show that the vectorU = (2, —1, —1) belongs to Span{(1,2,7),(2,5,1?],(—1,2,5)}. 7". [6pts) Answer True or False. Give reasons for your answer. (a) ||UxV||2+(U-V)2—{U-U)(V-V]=0. (b) The area of the triangle (ABC) equals 10, where A[1,1,0),B[0,1,1) and C(1,0,1). (c) (1R3, GB, 6)) is a vector space, where (sweaty, 2’) = (3+r’1y+y’.2+2’) and 09(ay.2) = {CLZCNZJ- (d) If X is a unit vector in ER“, and Y is any vector in 31‘5", then X is orthogonal toY—(X-YJX. “ CO Kuwait University Department of Mathematics and Computer science MATH 111, LINEAR ALGEBRA 30 July, 2009, 6—73me. Second Exam [Solutions]. Answer all questions. Calculators and mobile phones are NOT allowed. 1. (ths) Suppose “U” = 2 and “V” = 3. [s it possible that U-V = —7? (Justify your answer). . _ Luv 2 __7 _ No. Since cos|9 — —||U||||V|| s < 1. l 2 0 l 0 1 —2 l 2. (3pts) A: 0 I) 3 2 —l I] 0 l (a) Find the cofactor A11. A11 = (—1)2(3) = 3. (h) Evaluate |A| using cofactor expansion. A11 =(—1)2(3)1 A41 = {—1)5{—11) = 11- |A| —a-11A11 I a41A41 — (1N3) I i 1M“) — 8- (c) Use part (b) to evaluate éadjmjl. gum = (34 ladJKAN = (3412413 = %(—8)3 = —32- 3. (3pts) Show that adj (AB) = ndj(B]adj [A], where A and B are nonsinguiar n X :1 matrices. AB adj(AB) = |AB|I. Multiplying by (14-1): 141-1143 adj(AB} =A—1|A||B|I. Multiplying by (3—1): 3—13 adj(AB) = 34441—1 |A| |B| I. Since |1¢1|A_1 =ndj(A) and |B|B_1 = adj(B]. We get: adj(AB) = 1165(3) adj(A]. 1L (3pts) Find the point where the line through the points A(3, —4, —5) and 3(2, —3, l) crosses the plane 2x+y+z=7. U =AB =< —1,1,6 }_ Parametric equations: x —3 t; y — 4 I t; z— 5 613, t 5R. To find the required point: solve for t: 2(3 —t) + (—4 -- t] + (—5 +60 = T". Thus,6—2t—4+t—5+6t—T=D: i=2. The point is (1,—2,?]. 5. (4pts) (a) Determine whether the points Afl, l, 1) , 3(2, —1,3) and C(l], 3, 3] are on the same line. U = AB =<1,—2,2 >. Parametric equations: 17 = 1+t; y = 1— 2:; z = 1+ 26, t E 32. 0 belongs to this line if its coordinates satisify: 0 = 1 --t 3 = 1 — 2t 3 = 1-- 2:. This system has no solution (t = —1 and t = 1]. Thus, C is not on the line. (h) Determine whether the points P(—1,D,2),Q(1,2,D], R[—1,1,3) and SGML—:1) are on the same plane. PQ =< 2,2,—2 >, PR =<0,1,1:> : n =PQ xPR=< 4,—2,2 }_ l'[: 4(x+1)—2(y—D]+2(z—2)=0. . \ CheckifSisinl'I: 4(2+1)—2(0—0)+2(—4—2)=0: YES. 11$! .9 6. (Aims) (a) Letfl= {(33,152) 6313: 2:1:2y-I—z}. 1s 9 asubspace ofatg? 9 consists of all 3—vectors of the form (3:, 31,23 — 3;). (0,0,0) E Q: Q is non-empty. a] If U = (3,31,23 — y) E Q and U" = (3",y',23:" — y") E Q, then U—I—U" = (x,y,2r—y)—I—(I",y',2x’ yr) — (a: I say I y',2(:1: I 3:") (y I y')] H. ,3] If}: E 3% andU = (I, y,2J:—y) E Q, then kU = HI, y,2I—y] = [kr,ky,2kr—ky] E Q. (b) Show that the vector U = (2,—1,—1] belongs to Span“1,2,7],(2,5,17),[—1,2,5]}. Can we find real numbers c1,c>2,cg such that c1V1 -I— chg -I— {:ng = U. 1 2 —]. 2 2 5 2 —] Replacing 32 by R2 — 2R1 (EC Ra by R3 — 7R1: 7 17 5 —1 l 2 —1 2 l 2 —1 2 0 1 4 —-5 Replacing 33 by R3 — 3R2: 0 1 :1 —5 . 0 3 12 —15 0 0 0 0 The system has infinitely many solutions. 7. (fiptsj Answer True or False. Give reasons for your answer. (a) IIUxV||2+(U-V)2—(U-U)(V-V)=o. TRUE: IIUIIQII'Vllgsinfl59+IIUIIEIIVII"13015219—IIUIIQIIVII2 =0- [b] The area of the triangle (ABC) equals 10, where A[l,i,0},3[0,1,1) and C(1,0, l]. FALSE : AB 26. —1,0,1>.AC=< 0,—1, 1 >. AB x AC =< 1,1,1 >. AREA = %fi. (c) [333, EB, O) is a vector space, where [x,y,z) EB [I’,y’,z’] = (a: -|—:.":",y —I— y',z —I—z"] and c0 (3:,y, z) = (cr,2cy,cz]. FALSE : Since G G) [d (F) U) = c E) [d3,2dy,dz) = [ch,—iody,odz) and C(16) U = [dx,2cdy,cflz]. OR simply: 19 (1,2, 3) = (1,4,3) 72 (1,2,3). (d) If X is a unit vector in 3?“, and Y is any vector in 3%", then X is orthogonal toY—(X-Y)X. TRUE:X-(Y—(X-YJX)=X-Y—IIXII2X-Y20. OO 99 ...
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