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MATH201MT1 2007

# MATH201MT1 2007 - III-I Name Surname Student No Recitation...

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Unformatted text preview: III-I Name, Surname, Student No: Recitation Section: Math 201 Midterm I Duration: 90 minutes Part A: Short questions, 5 points each, no points without rea- soning I. Let A be a square matrix with the property A2 = A. Show that A = I or det(A) = 0. Sap/7052 #3,! GA; /A) #0. '77?“ A 7; max/‘55. lfAz=r4, Lem A’fAi A?) => :7 Ac; 1 II. Let T be a linear transformation from R2 in R3 and suppose that T((1,1)) = (1,0,1) and T((3, 2)) = (—1,1,2). Determine T((5, 4)) Explain how you obtained the value! Ti: 4: linear lmnsﬁimaﬂon/ C’onxymlﬁ T/X+y)= WNW” ﬁrak ><,y 619/. kéﬂ metre 77/5» 7(z/l’)+(§))= 277% 77rng = 2607* (22% (4') III. Let A be an n X n—matrix and suppose that det(A) = a. What is the determinant of —A. Explain! d“ “A” CW WM) = (-4JW/A)=(.4)"£ = z ifn/lreoen => ((01 f_£ n [-5 IV. Choose the correct answer (a) or (b) and explain: Let A, B be matrices such that the product AB is deﬁned. Suppose that the homogenous linear system An : 0 has inﬁnitely many solutions and Ba: 2 0 has a unique solutions. Then the linear system (ABMs = 0 has (a) a unique solution if 3*')’ 5’ mt“ “ST/“11h any, . rﬁo of Axfoo (b) inﬁnitely many solutions if Betsy />‘ C‘oﬂS/S‘fﬂl a 75 "f Ax - because: parlimbo— WI: 0%“? 2°00 ’f "I a 57””! mob” ‘ (AB)>< " 0 305 o nanlrl'viMSoKJr'on on! #2”ng lﬂf/Imléé many Sow»: Rm any) a We?» ye; Axgg 5mg Wal- 73x=y is garish-4n! fny). In purl/“cuter ifg 95 057W” maln'x 12% (3x ‘ fun/fut) 3%.. ﬁr aﬂ/ (Since 3x=0 ﬁasoné whirl/shale») Om! Men 648)): ﬂax; I.” "10% 30% Part B: I. Consider the linear system =yﬂﬂ¢0 COCO \$1 + x2 + 333 2.735 2373 + 334 2:55 2 1:1 + x2 + 2553 + 274 —— 32:5 213 —— 215 : 1. Write down the augmented matrix of the linear system. 2 points 2. Find the reduced row echelon form (rre f ) of the augmented matrix. 8 't I I I o z o R3434 0/0/21 02-3 R392 o/oI/ICIJ/Zg Ryzkzpmnsl IO 20 oozlza—eoczzlo"’ Douzom 0011/0 1/2/30 002020 002020 000-100 502020 000-200 11/020 _ 0 8—3‘2’?’ Lz—R’ 00/0/0152 o/olxg/lg (—0123 000/00 0000’00 000/00 000000 00000 000000 3. Find the solution set of the linear system. 4 points a ’(2 x3 “I: K: frOmI'r! o I I O o I o f o 0(Do I O 0 o [email protected] O O 0 O o O O 0 1706046154 Maxims/1 “I __/ X4 I 0 _ X: 0 xs‘f X11 0 I x, 0 11. Let T be the linear transformation from R3 in R3 given by T((x,y,2)) — (x+(k 2)z,:c : y : Zkz, a: 1 y+k22) for a real number k. (i) Find the standard matrix 5 points I o k—Z £73= l I 2k — l I u“ (ii) Find the values for k for which T is one to one respectively not one to one. 5 points -T— ,‘g ombmg=5 [T] if iﬂwrl;[email protected], or rrefof’fT] 121;! [0 if ‘35” 0 / 1&2 )L’& 0 / k+z 2— i I kl R3+R4 0/ k2+l(-2 0 0 k 1/ ,l I / ru—zNuz) =3 [7] 4:0 inm’jl‘“, lie. T}: onlooer )f can! 0% 432,. K24: {L—z) (1(+Z)#0 => 7/5 one/00’" if #12 ,‘f 1(=2 cv’lI=-‘Z. 775/701 one/cone (iii) For k : 1 determine [T ’1]. 6 points 10—1100 ,2 /0—/100 g /o—/Ioo A]? [IZOIOEz—iv 0134/0 559‘ 4,0 3.? .11100 Rer" 0/ ,l-J._.L lo—l/Oo g /00333 (0/3410 65—3) 0/0/0/ 00 I__(2{é-é ﬁZ—gez [—‘Ez’é’é' _4 //'/ = =/303 >27] 3-2”, (iv) For k = 1 determine three elementary matrices E1, E2, E3 such that [T] = E3E2E1U for an upper triangular matrix U . Put L = E3E2E1 such that [T] = LU. Which kind of matrix is L? 8 points 00—3 ' 4/ 3— 7- mg” loo) (o/f’g) 100 ; rceoare ,_ o. 010 méo'mﬁo”‘“a’m‘%m I \$4” 10/ j/o—Il ‘4 —7 —4 E 7:2 7"; ==>M=EE+IET1 =7 F; 45+; or, [T] [00 ,, _ =77 =WED-(3:?) 3 - EEzt”: 1/0 [00 7:4 3 7],! ’52.: (35 (/9): a [SaloMrsziaﬁguz'rmah’lx III. (1) Let u : (u1,u2),v : (711,112) be two vectors in R2 and suppose that the matrix satisﬁes AAT = I. (i) What can be said about the vectors u, v geometrically (2 observa— tons; a v “42* [422- uﬂmé ((1 V; a ,u u'v ‘- T A 2 4 4 = = _ AA = (w V2 512 V2) WNW: vaz+vf wv vov (I 52 =¢u.u=//u//z=4/- v-v=//V//2=// => //a//=//v//=// O . 2/ Lid/=0 => Uomf v are ofﬁogam! [700) (ii) Find the (unique) solution for u, v with U1,U27U1,1}2 Z O. V=(0//) {or “(ﬂany/t V: [40)] “a 6 points (2) Suppose that A,B are two 71 X n square matrices with det(A) = a, det(B) = ﬂ. Determine det(adj(AB)). (Explain your result.) 6 points one mm: {4/ ﬂ'gij) mgr 0“) => JOHN-’4‘; «£6 04) for a“ Myer/[\$6 ﬂxh mama” {2) 4a MA) = bndde) feral! 46/? A ' Mmzm [Am-male (3) cu m 8)= M/A)Jd(gj A/E Mame nxn'ma/ﬁag “A ,4 inmrﬂibe mm Moldy 1 ~\ V3 \\ [I Q 3. u ...
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