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Unformatted text preview: O. A02/002/01 . Use T.c. 16s MATH 2451 thqr02 Exam ble CalcUlus. 2016—0126
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assesseseeeeeeeseeeseee SMBQG was wé®® 0 (Last Name) 10 2016—0119 14: CftﬁvfﬁJ! €234" I xi sayN bibtgjf/JV'AO:§ O. AO2/OO2/02 0 1_. (1 points) Let c, .5 E R be nonzero, ,r Find the real eigenvalues or Show there are no . ",1 I? , 7 ,
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C +4 «a AL?“ Aﬁ/ )1: (HDLS 7i=d—/gg (b) Find the real eigenvalues or Show there are none. Ac [6 1 3“6 @an13 l; a; /r; i; b
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/ 63%;, RH" 3F . :r' t : if: (c) Find the real eigenvalues or Shaw there are none. I‘d—r. A [E i] i
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O 009 ©eeeeeeeeeo O a .. A02/002/0L . 5. (10 points) Evaluate. They all endup as “nice” expressions. You’ll see all of these later!
(3.) Easy. 1
det [003(9) —T 3mm] m (305 Q ragaJrémiérg‘gwe} : Y(40§79 l'grdjfxg )
sin(6) 1" 005(9) “’"' VGD§~Q+ r6?! N26, :2 w (b) Madame A Mr... * + cos(l9) —r sin(9) O 1...! I, W j 1 f + —r det sin(l9) T 005(9) 0 —r l # Vgl “’6 I. 2:? (‘Jzizi‘iy ' QN'E} l"‘i"'£‘~ﬁ'{7l “
f— + ; .O”   0m 1  . l ‘ r . . u "r I. “I NJ. A I”; I ’ —r r pm_  awe #kawl ragengw ‘1 ‘ L” A. I _ If \i Aim. WIN. (0 Her ﬁgure it out somewhere else and then neatly copy it here. Jr“ * [5mm (305(9) —p sin(¢) sin(6'] p cos[q15) 005(6)]
_ det p sin(¢) cos(6_] p cos(qb] sin(6) r,4/ _ ewe 0 —[email protected] 1 l
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01 (3.) Plot the parallelogram determined by Mt 61 and Mt Eg for the given value of t. ‘ l I F a _ O l ’ J 3. (10 points) Consider the matrix M: = [ ], known as a horizontal shear. / .—. .. ‘ _ " I _ .:‘
t— .sz D: E: Dig/LEII—J r . a /., _
It is very important to notice that det M3 = 1 which implies that Mt is area—preserving (all of your parallelograms should have the same area, which is interesting because you horizontally shear the unit square out to t = 1042
without changing its area). f @ onsider the matrix A = [63, 5], where ('1' and 5 are given in the plot. '
i. Determine the matrix R which will rigidly rotate the the indicated parallelogram such that the image of a: will sit on the positive r—axis. That is, the columns of RA = [R 5, R3] will determine the new, rotated
parallelogram. ii. Determine the matrix S which will horizontally shear RA into a rectangle. That is, columns of S RA will
moi/“‘1‘ Sé,determme a rectangle. ' If  _m____: JC’ WEE/)1
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/‘ x /— (Jr Procedures such as this are related to matrix factorizations which are varied and important. The purpose here
is to make sure you understand the associated geometry of what is happening and not just the (important, but
boring) steps of something like an L U factorization. ,L Grader Only J, 0 ea xa<’[email protected]@@©®@@® . 0. A02/002/05 Q 43}— 15;?" 4. (10 points) The shaded regions are ellipses centered at the origin.
(a) Find the matrix R which performs the following (mapping, and compute det R. 4be% if/
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Igo dose 3519 if'ﬂw‘a :! ;._.—— (b) Find the matrix L whichperfornls the following mapping, and compute det L. / r— ' ,a— , ‘ " Ji—t‘
X \. J: I_»_ ’7: _: 12$ a 6k [0? 2: v I i 3 C3 ’.:j{j' i‘“73 % i,» 3 =1; IFSu Li
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r f. . (c) Using the previous parts, ﬁnd the matrix M which performs the following mapping, and compute det M
W1! may: 61,1. .2 h‘ii‘ q = »  A ,fL—izsJ
,, ;_ t}
. El (d) What is the area of the tilted ellipse?_ f .. In, x I; P in;
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