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Unformatted text preview: Q. A01/014/01 0 Use T.C. 16s MATH 2451 thq_01 Mult Exam ble Calculus 1varia Course 15 min Duration 20160121 Due Date Instructor Net ID (First Name) SW wé W
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@[email protected]@m3§gg mu @ new @ @ © ® @J @ any .... \ i. ( “4L @ a @ mm.“ mw ® ® ® aw mm. @ ® ® ® O new 0 A: w ® ® @ mm ® ® ® ® I. \) a®@@@E®®@@©@®@@®@@®©a®e®C®®®®®®®®®
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[email protected]®©@@@@®@®©@®®@®@@®@@@@®@®®[email protected]®®®®@ M @ Ame ® as. new @ ® @ ® ® me ® E a @ ,9 my ® ® 0 ® ® ® @ my @n ® ® sen. (Last Name) 20160144 14: 15 .. A01/014/O2 . 1. (10 points) The unit circle is_;_ Wé MN IT, (Mi! Sl=ﬂnyﬂrt+f=d}' Iépdarga Simiﬂﬁﬁéf JF'HLT‘ (a) Show that 31 is not a subspace; of}? " ’din two points W7 (b) iAre theredany two points in 5'1 whose sum is again in 81? Find the conditions which i, 5 E 31 must satisfy so
“ that 5 + b E 31, and draw a sketch of a particular 6i and b which satisfy thes§__ conditions. _:_;__ _ *7— I {a
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(a) Suppose points A, B and C are vertices of a triangle. Find E + R: + a and provide a sketch.
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7:1 (b) Do the following lines intersect?
ﬁt) = (4, 5, #2) + (1,4, 1) t, 50:) : (3,1,—3) + (2,1,2)t=._
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€£g~4f§:ﬁ2k 9 ¢ Grader Only J, .0 A01/014/04 Q 3. (10 points) In this exercise, you’ll demonstrate the CauchySchwarz inequality in R” using a technique which is mo.
general than the proof provided in lecture. For any two vectors 13', u? E R“, lﬁtﬁl S ll’ﬁll llﬁill: with equality iff 13' or 1B is a scalar multiple of the other. (a) Show the statement is true if “E or u? is 6. (b) Explain why the function gilt) = “13+ trﬁHz is non—negative. éQi/c Nah} é :
(c) Show that the function is a quadratic in t by multiplyingout and collecting like terms until it is in the form
a t2 + M + c. (d) Use the quadratic discriminant b2 — 4 ac together with the previous two parts to conclude the inequality. The rest of the statement . . with equality iff . . .) will not be graded, but you should try to determine Why it is true
(but not on this sheet of paper).  \ 7 a: ._., __:_
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0 oo O .. A01/014/05 . =4. (10 points) (a) In lecture we saw that the diagonals of a parallelogram bisect each other. Show that the diagonals of a. parallel—
ogram are orthggial to each other iff the parallelogram is a rhombus. F OA/Eab/"ﬂl av Dlﬂéppﬂjé’iﬂéﬁwﬁgﬁﬂ £1.56“; Hip/é: 3.). ngﬁt> : a 5’7
'1. (b) Determine the matrix A which would map the indie . parallelogram onto the unit square. HDIM’? V ’F A f5? ‘ T1 HZ, _ o é/aé
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(a .. A01/014/06 . 5. (10 points) (a) Let R9 : 1R2 —) R2 denote the linear transformation which rotates a vector counterclockwise around the origin by
6. Recall that the matrix of R9 is given by: [Rs] = [$33 Tigil : Therefore RC3+¢) = Egg 0 R9, and since compulsitioh'of linear operators is given by matrix/multiplication, it must
be true that [R¢][R9] = [R(9+¢)]. Use this to fact to establish the following trigonometric identities. $5? c0509 + g6) = 005(9) cos(q5) — sin(l9_) sin(q§)
é; sin(6‘ + 915) = sin(6) cos(¢>) + 005(9) sin[¢)_ r’ "f: a l'; r '  ;:"_...__r" L
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:13?» _ _ — A, q , rm o >
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\__ (b) Let a, E e R2. Show: “EH2 “Eng .. (a . S)2 = (dens, 13])2
it; “1,619 It. 1122. ,i ‘
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“:Qidfirbib i u H / J. Grader Only J, O. A01/014/07 O 6. (10 points) Suppose 0:1
a: “2 A=[a,5=e1
033 kH
iﬁﬂ~1ﬁﬂ7A+43ﬁ +b5ﬁ C3 0"}? cﬁﬁ
Show that . J Momma 1.21.: A. (Bf7 52543": _' bl (:1
b = 62 ' det(A) = ya. (5' x a} Vow/mg ‘ [x ﬂéA CF" wag) 0—10 a; Hwy?) 5"?“ M 3 mi
wwwwwatr .m— “1.1“ a; I ’ “I;
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I a adv} C I 3 b“! 6‘! L34 L4 t: 3: 195.6: :dﬂgkuj?’ ‘03 ﬂ'5\’r 3 ‘9?
. E03 ﬁg? 7 . IN ‘ {’12) '/ f’r ‘ — I {#43721 _ 37:; a7: l V, O. A01/014/08 Q 7. (10 points) Additional questions which won’t be graded, but that you should know how to solve. 1. The product in R is cancellable: [a 75 O and a m : try] :> :1: 2 y. Show that the dot product in R”, n > 1 and cross product in R3 are not cancellable. 2. Showthat [(1%ﬁandae=agandsxe=sxg] 25:37. 3. The product in R is associative because (53 y z = :5 (y z) = m yz. Because it does not matter where the parentheses
are inserted, there is no chance of ambiguity and we usually drop the parentheses. Show that the cross product
is not associative by ﬁnding some vectors in R3 such that (53' x g) X :3 # i’ x [3} x ...
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